id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-22100 | Transpose of Outer Product | Let $\mathbf u = \tuple {u_1, u_2, \ldots, u_m}$ and $\mathbf v = \tuple {v_1, v_2, \ldots, v_n}$ be vectors.
Let $\mathbf u \otimes \mathbf v$ denote the outer product of $\mathbf u$ and $\mathbf v$.
Then:
:$\mathbf u \otimes \mathbf v = \paren {\mathbf v \otimes \mathbf u}^\intercal$
where ${}^\intercal$ denotes the ... | We have:
{{begin-eqn}}
{{eqn | l = \mathbf u \otimes \mathbf v
| r = <nowiki>\begin {bmatrix}
u_1 v_1 & u_1 v_2 & \dots & u_1 v_n \\
u_2 v_1 & u_2 v_2 & \dots & u_2 v_n \\
\vdots & \vdots & \ddots & \vdots \\
u_m v_1 & u_m v_2 & \dots & u_m v_n
\end {bmatrix}</nowiki>
| c = {{Defof|Outer Product}}
}}
{{eq... | Let $\mathbf u = \tuple {u_1, u_2, \ldots, u_m}$ and $\mathbf v = \tuple {v_1, v_2, \ldots, v_n}$ be [[Definition:Vector (Linear Algebra)|vectors]].
Let $\mathbf u \otimes \mathbf v$ denote the [[Definition:Outer Product|outer product]] of $\mathbf u$ and $\mathbf v$.
Then:
:$\mathbf u \otimes \mathbf v = \paren {\ma... | We have:
{{begin-eqn}}
{{eqn | l = \mathbf u \otimes \mathbf v
| r = <nowiki>\begin {bmatrix}
u_1 v_1 & u_1 v_2 & \dots & u_1 v_n \\
u_2 v_1 & u_2 v_2 & \dots & u_2 v_n \\
\vdots & \vdots & \ddots & \vdots \\
u_m v_1 & u_m v_2 & \dots & u_m v_n
\end {bmatrix}</nowiki>
| c = {{Defof|Outer Product}}
}}
{{e... | Transpose of Outer Product | https://proofwiki.org/wiki/Transpose_of_Outer_Product | https://proofwiki.org/wiki/Transpose_of_Outer_Product | [
"Outer Products",
"Transposes of Matrices"
] | [
"Definition:Vector/Linear Algebra",
"Definition:Outer Product",
"Definition:Transpose of Matrix"
] | [
"Category:Outer Products",
"Category:Transposes of Matrices"
] |
proofwiki-22101 | Subalgebra Generated by Inclusion and Conjugate is Everywhere Dense in Space of Continuous Functions on Compact Subset of Complex Numbers | Let $K \subseteq \C$ be compact.
Let $\map \CC K$ be the space of continuous functions on $K$.
Let $\iota : K \to \C$ be the inclusion mapping.
Let $\overline \iota : K \to \C$ be the complex conjugate.
Let $\AA$ be the subalgebra generated by $1$, $\iota$ and $\overline \iota$.
That is, by Explicit Form for Generated... | From Subalgebra Generated by Self-Adjoint Set is Self-Adjoint, $\AA$ is a $\ast$-subalgebra of $\map \CC K$.
We use the Complex-Valued Stone-Weierstrass Theorem: Compact Space.
We clearly have $1, \iota \in \AA$ by the definition of the generated algebra.
We have $\map \iota z = \map \iota w$ for $z, w \in K$ with $z \... | Let $K \subseteq \C$ be [[Definition:Compact Topological Space|compact]].
Let $\map \CC K$ be the [[Definition:Space of Continuous Functions on Compact Hausdorff Space|space of continuous functions on $K$]].
Let $\iota : K \to \C$ be the [[Definition:Inclusion Mapping|inclusion mapping]].
Let $\overline \iota : K \... | From [[Subalgebra Generated by Self-Adjoint Set is Self-Adjoint]], $\AA$ is a [[Definition:*-Subalgebra|$\ast$-subalgebra]] of $\map \CC K$.
We use the [[Stone-Weierstrass Theorem/Compact Space/Complex-Valued|Complex-Valued Stone-Weierstrass Theorem: Compact Space]].
We clearly have $1, \iota \in \AA$ by the definiti... | Subalgebra Generated by Inclusion and Conjugate is Everywhere Dense in Space of Continuous Functions on Compact Subset of Complex Numbers | https://proofwiki.org/wiki/Subalgebra_Generated_by_Inclusion_and_Conjugate_is_Everywhere_Dense_in_Space_of_Continuous_Functions_on_Compact_Subset_of_Complex_Numbers | https://proofwiki.org/wiki/Subalgebra_Generated_by_Inclusion_and_Conjugate_is_Everywhere_Dense_in_Space_of_Continuous_Functions_on_Compact_Subset_of_Complex_Numbers | [
"Space of Continuous Functions on Compact Hausdorff Space"
] | [
"Definition:Compact Topological Space",
"Definition:Space of Continuous Functions on Compact Hausdorff Space",
"Definition:Inclusion Mapping",
"Definition:Complex Conjugate",
"Definition:Generator of Algebra",
"Explicit Form for Generated Subalgebra",
"Definition:Everywhere Dense"
] | [
"Subalgebra Generated by Self-Adjoint Set is Self-Adjoint",
"Definition:*-Subalgebra",
"Stone-Weierstrass Theorem/Compact Space/Complex-Valued",
"Definition:Generator of Algebra",
"Definition:Mappings Separating Points",
"Stone-Weierstrass Theorem/Compact Space/Complex-Valued",
"Definition:Everywhere De... |
proofwiki-22102 | Spectral Mapping Theorem | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $x \in A$ be normal.
Let $\map {\sigma_A} x$ denote the spectrum of $x$ in $A$.
Let $\map \CC {\map {\sigma_A} x}$ be the space of continuous functions on $\map {\sigma_A} x$.
Let $f \in \map \CC {\map {\sigma_A} x}$.
Let $\map {\The... | Let $B \subseteq A$ be the $\text C^\ast$-algebra generated by $\set { {\mathbf 1}_A, x, x^\ast}$.
In the proof of Existence and Uniqueness of Continuous Functional Calculus, it is shown that $\Theta_x : \map \CC {\map {\sigma_A} x} \to B$ is an isometric unital $\ast$-algebra isomorphism.
Hence by Spectrum of Image of... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $x \in A$ be [[Definition:Normal Element of *-Algebra|normal]].
Let $\map {\sigma_A} x$ denote the [[Definition:Spectrum (Spectral Theory)/Unital Algebra|spectrum]] of ... | Let $B \subseteq A$ be the [[Definition:Generated C*-Algebra|$\text C^\ast$-algebra generated]] by $\set { {\mathbf 1}_A, x, x^\ast}$.
In the proof of [[Existence and Uniqueness of Continuous Functional Calculus]], it is shown that $\Theta_x : \map \CC {\map {\sigma_A} x} \to B$ is an [[Definition:Isometric Isomorphis... | Spectral Mapping Theorem/Proof 2 | https://proofwiki.org/wiki/Spectral_Mapping_Theorem | https://proofwiki.org/wiki/Spectral_Mapping_Theorem/Proof_2 | [
"Spectral Mapping Theorem",
"Continuous Functional Calculus",
"Spectral Mapping Theorem"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Normal Element of *-Algebra",
"Definition:Spectrum (Spectral Theory)/Unital Algebra",
"Definition:Space of Continuous Functions on Compact Hausdorff Space",
"Definition:Continuous Functional Calculus"
] | [
"Definition:Generated C*-Algebra",
"Existence and Uniqueness of Continuous Functional Calculus",
"Definition:Isometric Isomorphism",
"Definition:Unital Algebra Homomorphism",
"Definition:*-Algebra Isomorphism",
"Spectrum of Image of Element of Unital Algebra under Unital Algebra Homomorphism",
"Spectrum... |
proofwiki-22103 | Continuous Functional Calculus Commutes with Character on Generated C*-Subalgebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $x \in A$ be normal.
Let $\map {\sigma_A} x$ denote the spectrum of $x$ in $A$.
Let $\map \CC {\map {\sigma_A} x}$ be the space of continuous functions on $\map {\sigma_A} x$.
Let $f \in \map \CC {\map {\sigma_A} x}$.
Let $\map f x$ ... | From Spectrum of Element of Unital C*-Subalgebra of Unital C*-Algebra, we have:
:$\map {\sigma_A} {\map f x} = \map {\sigma_B} {\map f x}$
Let $\norm {\, \cdot \,}_\infty$ be the supremum norm on $\map {\sigma_B} x$.
From Subalgebra Generated by Inclusion and Conjugate is Everywhere Dense in Space of Continuous Functi... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $x \in A$ be [[Definition:Normal Element of *-Algebra|normal]].
Let $\map {\sigma_A} x$ denote the [[Definition:Spectrum (Spectral Theory)/Unital Algebra|spectrum]] of ... | From [[Spectrum of Element of Unital C*-Subalgebra of Unital C*-Algebra]], we have:
:$\map {\sigma_A} {\map f x} = \map {\sigma_B} {\map f x}$
Let $\norm {\, \cdot \,}_\infty$ be the [[Definition:Supremum Norm|supremum norm]] on $\map {\sigma_B} x$.
From [[Subalgebra Generated by Inclusion and Conjugate is Everywher... | Continuous Functional Calculus Commutes with Character on Generated C*-Subalgebra | https://proofwiki.org/wiki/Continuous_Functional_Calculus_Commutes_with_Character_on_Generated_C*-Subalgebra | https://proofwiki.org/wiki/Continuous_Functional_Calculus_Commutes_with_Character_on_Generated_C*-Subalgebra | [
"Continuous Functional Calculus",
"Characters (Banach Algebras)"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Normal Element of *-Algebra",
"Definition:Spectrum (Spectral Theory)/Unital Algebra",
"Definition:Space of Continuous Functions on Compact Hausdorff Space",
"Definition:Continuous Functional Calculus",
"Definition:Generated C*-Alge... | [
"Spectrum of Element of Unital C*-Subalgebra of Unital C*-Algebra",
"Definition:Supremum Norm",
"Subalgebra Generated by Inclusion and Conjugate is Everywhere Dense in Space of Continuous Functions on Compact Subset of Complex Numbers",
"Definition:Sequence",
"Existence and Uniqueness of Continuous Function... |
proofwiki-22104 | Continuous Functional Calculus obeys Composition | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $x \in A$ be normal.
Let $\map {\sigma_A} x$ denote the spectrum of $x$ in $A$.
Let $\map \CC {\map {\sigma_A} x}$ be the space of continuous functions on $\map {\sigma_A} x$.
Let $f \in \map \CC {\map {\sigma_A} x}$.
Let $g \in \map... | Note from Existence and Uniqueness of Continuous Functional Calculus that $\map f x$ is normal.
From the Spectral Mapping Theorem, we have:
:$f \sqbrk {\map {\sigma_A} x} = \map {\sigma_A} {\map f x}$
and hence $g \in \map \CC {\map {\sigma_A} {\map f x} }$.
We are hence assured that $\map g {\map f x}$ is well-defined... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $x \in A$ be [[Definition:Normal Element of *-Algebra|normal]].
Let $\map {\sigma_A} x$ denote the [[Definition:Spectrum (Spectral Theory)/Unital Algebra|spectrum]] of ... | Note from [[Existence and Uniqueness of Continuous Functional Calculus]] that $\map f x$ is [[Definition:Normal Element of *-Algebra|normal]].
From the [[Spectral Mapping Theorem]], we have:
:$f \sqbrk {\map {\sigma_A} x} = \map {\sigma_A} {\map f x}$
and hence $g \in \map \CC {\map {\sigma_A} {\map f x} }$.
We are h... | Continuous Functional Calculus obeys Composition | https://proofwiki.org/wiki/Continuous_Functional_Calculus_obeys_Composition | https://proofwiki.org/wiki/Continuous_Functional_Calculus_obeys_Composition | [
"Continuous Functional Calculus"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Normal Element of *-Algebra",
"Definition:Spectrum (Spectral Theory)/Unital Algebra",
"Definition:Space of Continuous Functions on Compact Hausdorff Space",
"Definition:Continuous Functional Calculus",
"Definition:Continuous Functi... | [
"Existence and Uniqueness of Continuous Functional Calculus",
"Definition:Normal Element of *-Algebra",
"Spectral Mapping Theorem",
"Definition:Generated C*-Algebra",
"Definition:Generated C*-Algebra",
"Existence and Uniqueness of Continuous Functional Calculus",
"C*-Algebra Generated by Commutative Sel... |
proofwiki-22105 | Normal Element of C*-Algebra is Hermitian iff Spectrum is Real | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $x \in A$ be normal.
Let $\map {\sigma_A} x$ denote the spectrum of $x$ in $A$.
Then $x$ is Hermitian {{iff}}:
:$\map {\sigma_A} x \subseteq \R$ | We first take $A$ unital. | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $x \in A$ be [[Definition:Normal Element of *-Algebra|normal]].
Let $\map {\sigma_A} x$ denote the [[Definition:Spectrum (Spectral Theory)/Unital Algebra|spectrum]] of $x$ in $A$.
Then $x$ is [[Definition:Hermi... | We first take $A$ [[Definition:Unital Banach Algebra|unital]]. | Normal Element of C*-Algebra is Hermitian iff Spectrum is Real | https://proofwiki.org/wiki/Normal_Element_of_C*-Algebra_is_Hermitian_iff_Spectrum_is_Real | https://proofwiki.org/wiki/Normal_Element_of_C*-Algebra_is_Hermitian_iff_Spectrum_is_Real | [
"Hermitian Elements of *-Algebras",
"C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Normal Element of *-Algebra",
"Definition:Spectrum (Spectral Theory)/Unital Algebra",
"Definition:Hermitian Element of *-Algebra"
] | [
"Definition:Unital Banach Algebra",
"Definition:Unital Banach Algebra"
] |
proofwiki-22106 | Normal Element of Unital C*-Algebra is Unitary iff Spectrum is Subset of Unit Circle | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $x \in A$ be normal.
Let $\map {\sigma_A} x$ denote the spectrum of $x$ in $A$.
Then $x$ is unitary {{iff}}:
:$\map {\sigma_A} x \subseteq \set {z \in \C : \cmod z = 1}$ | === Necessary Condition ===
Let $A' \subseteq A$ be the $\text C^\ast$-algebra generated by $\set { {\mathbf 1}_A, x}$.
By C*-Algebra Generated by Commutative Self-Adjoint Set is Commutative, $A'$ is commutative.
Let $\Phi_{A'}$ be the spectrum of $A'$.
From Spectrum of Element of Unital C*-Subalgebra of Unital C*-Alg... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $x \in A$ be [[Definition:Normal Element of *-Algebra|normal]].
Let $\map {\sigma_A} x$ denote the [[Definition:Spectrum (Spectral Theory)/Unital Algebra|spectrum]] of ... | === Necessary Condition ===
Let $A' \subseteq A$ be the [[Definition:Generated C*-Algebra|$\text C^\ast$-algebra generated]] by $\set { {\mathbf 1}_A, x}$.
By [[C*-Algebra Generated by Commutative Self-Adjoint Set is Commutative]], $A'$ is [[Definition:Commutative Algebra (Abstract Algebra)|commutative]].
Let $\Phi... | Normal Element of Unital C*-Algebra is Unitary iff Spectrum is Subset of Unit Circle | https://proofwiki.org/wiki/Normal_Element_of_Unital_C*-Algebra_is_Unitary_iff_Spectrum_is_Subset_of_Unit_Circle | https://proofwiki.org/wiki/Normal_Element_of_Unital_C*-Algebra_is_Unitary_iff_Spectrum_is_Subset_of_Unit_Circle | [
"Unitary Elements of Unital *-Algebras",
"C*-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Normal Element of *-Algebra",
"Definition:Spectrum (Spectral Theory)/Unital Algebra",
"Definition:Unitary Element of Unital *-Algebra"
] | [
"Definition:Generated C*-Algebra",
"C*-Algebra Generated by Commutative Self-Adjoint Set is Commutative",
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Spectrum of Banach Algebra",
"Spectrum of Element of Unital C*-Subalgebra of Unital C*-Algebra",
"Spectrum of Element of Unital Commuta... |
proofwiki-22107 | Existence of Generated C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $S \subseteq A$.
Then there exists a $\text C^\ast$-subalgebra $B \subseteq A$ containing $S$ which is $\subseteq$-minimal among $\text C^\ast$-subalgebras with this property. | Let $\SS$ be the set of $\text C^\ast$-subalgebras $C \subseteq A$ containing $S$.
We have $A \in \SS$ and hence $\SS \ne \O$.
Let $B = \bigcap \SS$.
From Intersection of C*-Subalgebras is C*-Subalgebra, $B$ is a $\text C^\ast$-algebra.
Further, since $S \subseteq C$ for each $C \in \SS$, we have $S \subseteq B$.
We n... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $S \subseteq A$.
Then there exists a [[Definition:C*-Subalgebra|$\text C^\ast$-subalgebra]] $B \subseteq A$ containing $S$ which is [[Definition:Minimal Element|$\subseteq$-minimal]] among [[Definition:C*-Subalge... | Let $\SS$ be the [[Definition:Set|set]] of [[Definition:C*-Subalgebra|$\text C^\ast$-subalgebras]] $C \subseteq A$ containing $S$.
We have $A \in \SS$ and hence $\SS \ne \O$.
Let $B = \bigcap \SS$.
From [[Intersection of C*-Subalgebras is C*-Subalgebra]], $B$ is a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
F... | Existence of Generated C*-Algebra | https://proofwiki.org/wiki/Existence_of_Generated_C*-Algebra | https://proofwiki.org/wiki/Existence_of_Generated_C*-Algebra | [
"C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:C*-Subalgebra",
"Definition:Minimal/Element",
"Definition:C*-Subalgebra"
] | [
"Definition:Set",
"Definition:C*-Subalgebra",
"Intersection of C*-Subalgebras is C*-Subalgebra",
"Definition:C*-Algebra",
"Definition:Minimal/Element",
"Definition:C*-Subalgebra",
"Intersection is Subset",
"Definition:Minimal/Element",
"Category:C*-Algebras"
] |
proofwiki-22108 | Unital C*-Algebra is Unital Banach Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra that is unital as an algebra, with identity element ${\mathbf 1}_A \ne {\mathbf 0}_A$.
Then $A$ is unital as a Banach algebra. | We have:
{{begin-eqn}}
{{eqn | l = \norm { {\mathbf 1}_A}^2
| r = \norm { {\mathbf 1}_A {\mathbf 1}_A^\ast}
| c = {{Defof|C*-Algebra}}
}}
{{eqn | r = \norm { {\mathbf 1}_A^2}
| c = Identity Element in Unital *-Algebra is Hermitian
}}
{{eqn | r = \norm { {\mathbf 1}_A}
}}
{{end-eqn}}
From {{NormAxiomVector|1}}, w... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]] that is [[Definition:Unital Algebra|unital as an algebra]], with [[Definition:Identity Element|identity element]] ${\mathbf 1}_A \ne {\mathbf 0}_A$.
Then $A$ is [[Definition:Unital Banach Algebra|unital as a Banach alge... | We have:
{{begin-eqn}}
{{eqn | l = \norm { {\mathbf 1}_A}^2
| r = \norm { {\mathbf 1}_A {\mathbf 1}_A^\ast}
| c = {{Defof|C*-Algebra}}
}}
{{eqn | r = \norm { {\mathbf 1}_A^2}
| c = [[Identity Element in Unital *-Algebra is Hermitian]]
}}
{{eqn | r = \norm { {\mathbf 1}_A}
}}
{{end-eqn}}
From {{NormAxiomVector|1... | Unital C*-Algebra is Unital Banach Algebra | https://proofwiki.org/wiki/Unital_C*-Algebra_is_Unital_Banach_Algebra | https://proofwiki.org/wiki/Unital_C*-Algebra_is_Unital_Banach_Algebra | [
"Unital Banach Algebras",
"C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Unital Algebra",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Unital Banach Algebra"
] | [
"Identity Element in Unital *-Algebra is Hermitian"
] |
proofwiki-22109 | Surface Area of Cuboid | Let $\CC$ be a cuboid whose edges are of length $a$, $b$ and $c$.
The surface area $S$ of $\CC$ is given as:
:$S = 2 \paren {a b + b c + a c}$ | Each of the faces of $\CC$ are rectangles:
:$2$ of these faces are adjacent to edges of length $a$ and $b$
:$2$ of these faces are adjacent to edges of length $b$ and $c$
:$2$ of these faces are adjacent to edges of length $a$ and $c$.
Hence from Area of Rectangle:
:there are $2$ faces with area $a b$
:there are $2$ fa... | Let $\CC$ be a [[Definition:Cuboid|cuboid]] whose [[Definition:Edge of Polyhedron|edges]] are of [[Definition:Length of Line|length]] $a$, $b$ and $c$.
The [[Definition:Surface Area|surface area]] $S$ of $\CC$ is given as:
:$S = 2 \paren {a b + b c + a c}$ | Each of the [[Definition:Face of Polyhedron|faces]] of $\CC$ are [[Definition:Rectangle|rectangles]]:
:$2$ of these [[Definition:Face of Polyhedron|faces]] are [[Definition:Adjacent Face to Edge|adjacent]] to [[Definition:Edge of Polyhedron|edges]] of [[Definition:Length of Line|length]] $a$ and $b$
:$2$ of these [[Def... | Surface Area of Cuboid | https://proofwiki.org/wiki/Surface_Area_of_Cuboid | https://proofwiki.org/wiki/Surface_Area_of_Cuboid | [
"Cuboids",
"Area Formulas"
] | [
"Definition:Cuboid",
"Definition:Polyhedron/Edge",
"Definition:Linear Measure/Length",
"Definition:Surface Area"
] | [
"Definition:Polyhedron/Face",
"Definition:Quadrilateral/Rectangle",
"Definition:Polyhedron/Face",
"Definition:Polyhedron/Adjacent/Face to Edge",
"Definition:Polyhedron/Edge",
"Definition:Linear Measure/Length",
"Definition:Polyhedron/Face",
"Definition:Polyhedron/Adjacent/Face to Edge",
"Definition:... |
proofwiki-22110 | Intersection of *-Subalgebras is *-Subalgebra | Let $\tuple {A, \ast}$ be a $\ast$-algebra over $\C$.
Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be an $I$-indexed family of $\ast$-subalgebras of $A$.
Let:
:$\ds B = \bigcap_{\alpha \mathop \in I} A_\alpha$
Then $B$ is a $\ast$-subalgebra of $A$. | From Intersection of Subalgebras is Subalgebra, $B$ is a subalgebra of $A$.
It remains to show that for each $x \in B$ we have $x^\ast \in B$.
Let $x \in B$.
Then $x \in A_\alpha$ for each $\alpha \in I$.
Since each $A_\alpha$ is a $\ast$-subalgebra, we have $x^\ast \in A_\alpha$ for each $\alpha \in I$.
Hence we hav... | Let $\tuple {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]] over $\C$.
Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|$I$-indexed family]] of [[Definition:*-Subalgebra|$\ast$-subalgebras]] of $A$.
Let:
:$\ds B = \bigcap_{\alpha \mathop \in I} A_\alpha$
Then $B$ is a [[Defi... | From [[Intersection of Subalgebras is Subalgebra]], $B$ is a [[Definition:Subalgebra|subalgebra]] of $A$.
It remains to show that for each $x \in B$ we have $x^\ast \in B$.
Let $x \in B$.
Then $x \in A_\alpha$ for each $\alpha \in I$.
Since each $A_\alpha$ is a [[Definition:*-Subalgebra|$\ast$-subalgebra]], we hav... | Intersection of *-Subalgebras is *-Subalgebra | https://proofwiki.org/wiki/Intersection_of_*-Subalgebras_is_*-Subalgebra | https://proofwiki.org/wiki/Intersection_of_*-Subalgebras_is_*-Subalgebra | [
"*-Algebras"
] | [
"Definition:*-Algebra",
"Definition:Indexing Set/Family",
"Definition:*-Subalgebra",
"Definition:*-Subalgebra"
] | [
"Intersection of Subalgebras is Subalgebra",
"Definition:Subalgebra",
"Definition:*-Subalgebra",
"Category:*-Algebras"
] |
proofwiki-22111 | Intersection of C*-Subalgebras is C*-Subalgebra | Let $\tuple {A, \norm {\, \cdot \,}, \ast}$ be a $\text C^\ast$-algebra.
Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be an $I$-indexed family of $\text C^\ast$-subalgebras of $A$.
Let:
:$\ds B = \bigcap_{\alpha \mathop \in I} A_\alpha$
Then $B$ is a $\text C^\ast$-subalgebra of $A$. | From Intersection of *-Subalgebras is *-Subalgebra, $B$ is a $\ast$-subalgebra of $A$.
By definition, $A_\alpha$ is a complete metric subspace of $A$ for each $\alpha \in I$.
Hence from Subspace of Complete Metric Space is Closed iff Complete, $A_\alpha$ is closed in $A$.
Hence as the intersection of closed sets, $B$ ... | Let $\tuple {A, \norm {\, \cdot \,}, \ast}$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|$I$-indexed family]] of [[Definition:C*-Subalgebra|$\text C^\ast$-subalgebras]] of $A$.
Let:
:$\ds B = \bigcap_{\alpha \mathop \in I} A_\... | From [[Intersection of *-Subalgebras is *-Subalgebra]], $B$ is a [[Definition:*-Subalgebra|$\ast$-subalgebra]] of $A$.
By definition, $A_\alpha$ is a [[Definition:Complete Metric Space|complete]] [[Definition:Metric Subspace|metric subspace]] of $A$ for each $\alpha \in I$.
Hence from [[Subspace of Complete Metric S... | Intersection of C*-Subalgebras is C*-Subalgebra | https://proofwiki.org/wiki/Intersection_of_C*-Subalgebras_is_C*-Subalgebra | https://proofwiki.org/wiki/Intersection_of_C*-Subalgebras_is_C*-Subalgebra | [
"C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Indexing Set/Family",
"Definition:C*-Subalgebra",
"Definition:C*-Subalgebra"
] | [
"Intersection of *-Subalgebras is *-Subalgebra",
"Definition:*-Subalgebra",
"Definition:Complete Metric Space",
"Definition:Metric Subspace",
"Subspace of Complete Metric Space is Closed iff Complete",
"Definition:Closed Set",
"Definition:Set Intersection",
"Definition:Closed Set",
"Definition:Close... |
proofwiki-22112 | Continuous Function applied to Normal Element of Unital C*-Algebra is Positive iff Function is Non-Negative | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $x \in A$ be normal.
Let $\map {\sigma_A} x$ denote the spectrum of $x$ in $A$.
Let $f : \map {\sigma_A} x \to \C$ be continuous.
Let $\map {\Theta_x} f = \map f x$ be the continuous functional calculus of $x$ applied to $f$.
Then $... | From the definition of a positive element, we have that $\map f x$ is positive {{iff}} Hermitian and $\map {\sigma_A} {\map f x} \subseteq \hointr 0 \infty$.
Note that $\map {\sigma_A} {\map f x} \subseteq \hointr 0 \infty$ implies that $\map {\sigma_A} {\map f x} \subseteq \R$.
From Normal Element of C*-Algebra is H... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $x \in A$ be [[Definition:Normal Element of *-Algebra|normal]].
Let $\map {\sigma_A} x$ denote the [[Definition:Spectrum (Spectral Theory)/Unital Algebra|spectrum]] of ... | From the definition of a [[Definition:Positive Element of C*-Algebra|positive element]], we have that $\map f x$ is [[Definition:Positive Element of C*-Algebra|positive]] {{iff}} [[Definition:Hermitian Element of *-Algebra|Hermitian]] and $\map {\sigma_A} {\map f x} \subseteq \hointr 0 \infty$.
Note that $\map {\sigm... | Continuous Function applied to Normal Element of Unital C*-Algebra is Positive iff Function is Non-Negative | https://proofwiki.org/wiki/Continuous_Function_applied_to_Normal_Element_of_Unital_C*-Algebra_is_Positive_iff_Function_is_Non-Negative | https://proofwiki.org/wiki/Continuous_Function_applied_to_Normal_Element_of_Unital_C*-Algebra_is_Positive_iff_Function_is_Non-Negative | [
"Positive Elements of C*-Algebras",
"Continuous Functional Calculus",
"Continuous Function applied to Normal Element of Unital C*-Algebra is Positive iff Function is Non-Negative"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Normal Element of *-Algebra",
"Definition:Spectrum (Spectral Theory)/Unital Algebra",
"Definition:Continuous Function",
"Definition:Continuous Functional Calculus",
"Definition:Positive Element of C*-Algebra"
] | [
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Hermitian Element of *-Algebra",
"Normal Element of C*-Algebra is Hermitian iff Spectrum is Real",
"Definition:Hermitian Element of *-Algebra",
"Definition:Positive Element of C*-Algebra",
"Spectral Map... |
proofwiki-22113 | Existence and Uniqueness of Positive Nth Root of Positive Element of C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $x \in A$ be positive.
Let $n \ge 2$.
Then there exists a unique positive element $y \in A$ such that $x = y^n$. | We first take $A$ to be unital.
Let $B$ be a commutative unital $\text C^\ast$-subalgebra of $A$ with $x \in B$.
We show the existence and uniqueness of positive $y \in B$ (''not'' in the entirety of $A$ yet) such that $x = y^n$.
From Spectrum of Element of Unital C*-Subalgebra of Unital C*-Algebra we have $\map {\sig... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $x \in A$ be [[Definition:Positive Element of C*-Algebra|positive]].
Let $n \ge 2$.
Then there exists a unique [[Definition:Positive Element of C*-Algebra|positive element]] $y \in A$ such that $x = y^n$. | We first take $A$ to be [[Definition:Unital Banach Algebra|unital]].
Let $B$ be a [[Definition:Commutative Algebra (Abstract Algebra)|commutative]] [[Definition:Unital Subalgebra|unital]] [[Definition:C*-Subalgebra|$\text C^\ast$-subalgebra]] of $A$ with $x \in B$.
We show the existence and uniqueness of [[Definition... | Existence and Uniqueness of Positive Nth Root of Positive Element of C*-Algebra | https://proofwiki.org/wiki/Existence_and_Uniqueness_of_Positive_Nth_Root_of_Positive_Element_of_C*-Algebra | https://proofwiki.org/wiki/Existence_and_Uniqueness_of_Positive_Nth_Root_of_Positive_Element_of_C*-Algebra | [
"Positive Elements of C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Element of C*-Algebra"
] | [
"Definition:Unital Banach Algebra",
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Unital Subalgebra",
"Definition:C*-Subalgebra",
"Definition:Positive Element of C*-Algebra",
"Spectrum of Element of Unital C*-Subalgebra of Unital C*-Algebra",
"Definition:Spectrum of Banach Algebra",
... |
proofwiki-22114 | Hermitian Element of Unital C*-Algebra is Linear Combination of Two Unitary Elements | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $x \in A$ be Hermitian.
Then there exists unitary elements $u, v \in A$ and $\alpha \ge 0$ such that:
:$x = \alpha \paren {u + v}$ | If $x = {\mathbf 0}_A$, then we can take $\alpha = 0$ and $u = v = {\mathbf 1}_A$.
Take $x \ne {\mathbf 0}_A$.
First take $\norm x \le 1$.
From Spectral Radius of Normal Element of C*-Algebra Equal to Norm, we have:
:$\norm x = \sup \set {\cmod \lambda : \lambda \in \map {\sigma_A} x} \le 1$
where $\map {\sigma_A} x$ ... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $x \in A$ be [[Definition:Hermitian Element of *-Algebra|Hermitian]].
Then there exists [[Definition:Unitary Element of Unital *-Algebra|unitary elements]] $u, v \in A... | If $x = {\mathbf 0}_A$, then we can take $\alpha = 0$ and $u = v = {\mathbf 1}_A$.
Take $x \ne {\mathbf 0}_A$.
First take $\norm x \le 1$.
From [[Spectral Radius of Normal Element of C*-Algebra Equal to Norm]], we have:
:$\norm x = \sup \set {\cmod \lambda : \lambda \in \map {\sigma_A} x} \le 1$
where $\map {\sigma... | Hermitian Element of Unital C*-Algebra is Linear Combination of Two Unitary Elements | https://proofwiki.org/wiki/Hermitian_Element_of_Unital_C*-Algebra_is_Linear_Combination_of_Two_Unitary_Elements | https://proofwiki.org/wiki/Hermitian_Element_of_Unital_C*-Algebra_is_Linear_Combination_of_Two_Unitary_Elements | [
"Hermitian Elements of *-Algebras",
"Unitary Elements of Unital *-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Hermitian Element of *-Algebra",
"Definition:Unitary Element of Unital *-Algebra"
] | [
"Spectral Radius of Normal Element of C*-Algebra Equal to Norm",
"Definition:Spectrum (Spectral Theory)",
"Spectrum of Hermitian Element in Unital C*-Algebra is Real",
"Spectral Mapping Theorem for Polynomials",
"Star of Product of Elements in *-Algebra",
"Identity Element in Unital *-Algebra is Hermitian... |
proofwiki-22115 | Equivalence of Definitions of Frame Isomorphism | Let $L_1 = \struct {S_1, \preceq_1}$ and $L_2 = \struct {S_2, \preceq_2}$ be frames.
{{TFAE|def=Frame Isomorphism}}
=== Definition 1 ===
{{:Definition:Frame Isomorphism/Definition 1}}
=== Definition 2 ===
{{:Definition:Frame Isomorphism/Definition 2}}
=== Definition 3 ===
{{:Definition:Frame Isomorphism/Definition 3}}
... | By definition of frame:
:$L_1$ and $L_2$ are complete lattices
From Equivalence of Definitions of Complete Lattice Isomorphism definitions $2$, $3$ and $4$ are equivalent. | Let $L_1 = \struct {S_1, \preceq_1}$ and $L_2 = \struct {S_2, \preceq_2}$ be [[Definition:Frame (Lattice Theory)|frames]].
{{TFAE|def=Frame Isomorphism}}
=== [[Definition:Frame Isomorphism/Definition 1|Definition 1]] ===
{{:Definition:Frame Isomorphism/Definition 1}}
=== [[Definition:Frame Isomorphism/Definition 2... | By definition of [[Definition:Frame (Lattice Theory)|frame]]:
:$L_1$ and $L_2$ are [[Definition:Complete Lattice|complete lattices]]
From [[Equivalence of Definitions of Complete Lattice Isomorphism]] definitions $2$, $3$ and $4$ are equivalent. | Equivalence of Definitions of Frame Isomorphism | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Frame_Isomorphism | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Frame_Isomorphism | [
"Frame Isomorphisms",
"Equivalence of Definitions of Frame Isomorphism"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Frame Isomorphism/Definition 1",
"Definition:Frame Isomorphism/Definition 2",
"Definition:Frame Isomorphism/Definition 3",
"Definition:Frame Isomorphism/Definition 4"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Complete Lattice",
"Equivalence of Definitions of Complete Lattice Isomorphism"
] |
proofwiki-22116 | Frame Isomorphism is Isomorphism in Category Frm | Let $\mathbf{Frm}$ denote the category of frames.
Let $\phi : L_1 \to L_2$ be a morphism of $\mathbf{Frm}$.
Then:
:$\phi$ is an isomorphism of $\mathbf{Frm}$ {{iff}} $\phi$ is a complete lattice isomorphsm | Let $L_1$ and $L_2$ be the frames $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ respectively.
By definition of category of frames:
:$\phi$ is a frame homomorphisms
By definition of an isomorphism:
:$\phi$ is an isomorphism of $\mathbf{Frm}$
{{iff}}:
:$(1):$ there exists a morphism of $\psi : L_2 \... | Let $\mathbf{Frm}$ denote the [[Definition:Category of Frames|category of frames]].
Let $\phi : L_1 \to L_2$ be a [[Definition:Morphism|morphism]] of $\mathbf{Frm}$.
Then:
:$\phi$ is an [[Definition:Isomorphism (Category Theory)|isomorphism]] of $\mathbf{Frm}$ {{iff}} $\phi$ is a [[Definition:Complete Lattice Isomo... | Let $L_1$ and $L_2$ be the [[Definition:Frame (Lattice Theory)|frames]] $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ respectively.
By definition of [[Definition:Category of Frames|category of frames]]:
:$\phi$ is a [[Definition:Frame Homomorphism|frame homomorphisms]]
By definition of an [[De... | Frame Isomorphism is Isomorphism in Category Frm | https://proofwiki.org/wiki/Frame_Isomorphism_is_Isomorphism_in_Category_Frm | https://proofwiki.org/wiki/Frame_Isomorphism_is_Isomorphism_in_Category_Frm | [
"Category of Frames (Lattice Theory)"
] | [
"Definition:Category of Frames",
"Definition:Morphism",
"Definition:Isomorphism (Category Theory)",
"Definition:Complete Lattice Isomorphism"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Category of Frames",
"Definition:Frame Homomorphism",
"Definition:Isomorphism (Category Theory)",
"Definition:Isomorphism (Category Theory)",
"Definition:Morphism",
"Definition:Identity Morphism",
"Definition:Category of Frames",
"Definition:Frame Hom... |
proofwiki-22117 | Monomorphism iff Epimorphism in Dual Category | Let $\mathbf C$ be a metacategory.
Let $\mathbf C^{\operatorname{op}}$ be the dual category of $\mathbf C$.
Let $f$ be a morphism of $\mathbf C$.
Then $f$ is a monomorphism in $\mathbf C$ {{iff}} $f^{\operatorname{op}}$ is an epimorphism in $\mathbf C^{\operatorname{op}}$. | === Necessary Condition ===
Let $f$ be a monomorphism in $\mathbf C$.
Let $g^{\operatorname{op}}$ and $h^{\operatorname{op}}$ be morphisms of $\mathbf C^{\operatorname{op}}$:
:the compositions $g^{\operatorname{op}} \circ f^{\operatorname{op}}$ and $h^{\operatorname{op}} \circ f^{\operatorname{op}}$ are defined
We have... | Let $\mathbf C$ be a [[Definition:Metacategory|metacategory]].
Let $\mathbf C^{\operatorname{op}}$ be the [[Definition:Dual Category|dual category]] of $\mathbf C$.
Let $f$ be a [[Definition:Morphism (Category Theory)|morphism]] of $\mathbf C$.
Then $f$ is a [[Definition:Monomorphism (Category Theory)|monomorphism]... | === Necessary Condition ===
Let $f$ be a [[Definition:Monomorphism (Category Theory)|monomorphism]] in $\mathbf C$.
Let $g^{\operatorname{op}}$ and $h^{\operatorname{op}}$ be [[Definition:Morphism (Category Theory)|morphisms]] of $\mathbf C^{\operatorname{op}}$:
:the [[Definition:Composition of Morphisms|composition... | Monomorphism iff Epimorphism in Dual Category | https://proofwiki.org/wiki/Monomorphism_iff_Epimorphism_in_Dual_Category | https://proofwiki.org/wiki/Monomorphism_iff_Epimorphism_in_Dual_Category | [
"Epimorphisms (Category Theory)",
"Monomorphisms (Category Theory)",
"Dual Categories"
] | [
"Definition:Metacategory",
"Definition:Dual Category",
"Definition:Morphism",
"Definition:Monomorphism (Category Theory)",
"Definition:Epimorphism (Category Theory)"
] | [
"Definition:Monomorphism (Category Theory)",
"Definition:Morphism",
"Definition:Composition of Morphisms",
"Definition:Dual Category",
"Definition:Epimorphism (Category Theory)",
"Definition:Epimorphism (Category Theory)",
"Definition:Monomorphism (Category Theory)",
"Definition:Morphism",
"Definiti... |
proofwiki-22118 | Epimorphism iff Monomorphism in Dual Category | Let $\mathbf C$ be a metacategory.
Let $\mathbf C^{\operatorname{op}}$ be the dual category of $\mathbf C$.
Let $f$ be an morphism of $\mathbf C$.
Then $f$ is a epimorphism in $\mathbf C$ {{iff}} $f^{\operatorname{op}}$ is a monomorphism in $\mathbf C^{\operatorname{op}}$. | === Necessary Condition ===
Let $f$ be an epimorphism in $\mathbf C$.
Let $g^{\operatorname{op}}$ and $h^{\operatorname{op}}$ be morphisms of $\mathbf C^{\operatorname{op}}$:
:the compositions $f^{\operatorname{op}} \circ g^{\operatorname{op}}$ and $f^{\operatorname{op}} \circ h^{\operatorname{op}}$ are defined
We have... | Let $\mathbf C$ be a [[Definition:Metacategory|metacategory]].
Let $\mathbf C^{\operatorname{op}}$ be the [[Definition:Dual Category|dual category]] of $\mathbf C$.
Let $f$ be an [[Definition:Morphism (Category Theory)|morphism]] of $\mathbf C$.
Then $f$ is a [[Definition:Epimorphism (Category Theory)|epimorphism]]... | === Necessary Condition ===
Let $f$ be an [[Definition:Epimorphism (Category Theory)|epimorphism]] in $\mathbf C$.
Let $g^{\operatorname{op}}$ and $h^{\operatorname{op}}$ be [[Definition:Morphism (Category Theory)|morphisms]] of $\mathbf C^{\operatorname{op}}$:
:the [[Definition:Composition of Morphisms|compositions... | Epimorphism iff Monomorphism in Dual Category | https://proofwiki.org/wiki/Epimorphism_iff_Monomorphism_in_Dual_Category | https://proofwiki.org/wiki/Epimorphism_iff_Monomorphism_in_Dual_Category | [
"Epimorphisms (Category Theory)",
"Monomorphisms (Category Theory)",
"Dual Categories"
] | [
"Definition:Metacategory",
"Definition:Dual Category",
"Definition:Morphism",
"Definition:Epimorphism (Category Theory)",
"Definition:Monomorphism (Category Theory)"
] | [
"Definition:Epimorphism (Category Theory)",
"Definition:Morphism",
"Definition:Composition of Morphisms",
"Definition:Dual Category",
"Definition:Monomorphism (Category Theory)",
"Definition:Monomorphism (Category Theory)",
"Definition:Epimorphism (Category Theory)",
"Definition:Morphism",
"Definiti... |
proofwiki-22119 | Isomorphism iff Isomorphism in Dual Category | Let $\mathbf C$ be a metacategory.
Let $\mathbf C^{\operatorname{op}}$ be the dual category of $\mathbf C$.
Let $f$ be a morphism of $\mathbf C$.
Then $f$ is an isomorphism in $\mathbf C$ {{iff}} $f^{\operatorname{op}}$ is an isomorphism in $\mathbf C^{\operatorname{op}}$. | === Necessary Condition ===
Let $f: C_1 \to C_2$ be an isomorphism in $\mathbf C$.
Let $g: C_2 \to C_1$ be the inverse morphism of $f$ in $\mathbf C$.
By definition of inverse morphism:
:$g \circ f = \operatorname{id}_{C_1}$
:$f \circ g = \operatorname{id}_{C_2}$
where $\operatorname{id}_{C_1}, \operatorname{id}_{C_2}$... | Let $\mathbf C$ be a [[Definition:Metacategory|metacategory]].
Let $\mathbf C^{\operatorname{op}}$ be the [[Definition:Dual Category|dual category]] of $\mathbf C$.
Let $f$ be a [[Definition:Morphism (Category Theory)|morphism]] of $\mathbf C$.
Then $f$ is an [[Definition:Isomorphism (Category Theory)|isomorphism]]... | === Necessary Condition ===
Let $f: C_1 \to C_2$ be an [[Definition:Isomorphism (Category Theory)|isomorphism]] in $\mathbf C$.
Let $g: C_2 \to C_1$ be the [[Definition:Inverse Morphism|inverse morphism]] of $f$ in $\mathbf C$.
By definition of [[Definition:Inverse Morphism|inverse morphism]]:
:$g \circ f = \opera... | Isomorphism iff Isomorphism in Dual Category | https://proofwiki.org/wiki/Isomorphism_iff_Isomorphism_in_Dual_Category | https://proofwiki.org/wiki/Isomorphism_iff_Isomorphism_in_Dual_Category | [
"Isomorphisms (Category Theory)",
"Dual Categories"
] | [
"Definition:Metacategory",
"Definition:Dual Category",
"Definition:Morphism",
"Definition:Isomorphism (Category Theory)",
"Definition:Isomorphism (Category Theory)"
] | [
"Definition:Isomorphism (Category Theory)",
"Definition:Inverse Morphism",
"Definition:Inverse Morphism",
"Definition:Identity Morphism",
"Definition:Dual Category",
"Definition:Dual Category",
"Definition:Isomorphism (Category Theory)",
"Definition:Isomorphism (Category Theory)",
"Definition:Isomor... |
proofwiki-22120 | Non-Negative Multiple of Positive Element of C*-Algebra is Positive | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $x \in A$ be positive.
Let $\alpha \ge 0$.
Then $\alpha x$ is positive. | Let $\sigma_A$ denote the spectrum in $A$.
Since $x$ is positive, we have:
:$\map {\sigma_A} x \subseteq \hointr 0 \infty$
From Spectral Mapping Theorem for Polynomials, we have:
:$\map {\sigma_A} {\alpha x} = \set {\alpha t : t \in \map {\sigma_A} x}$
Since $t \in \map {\sigma_A} x$ has $t \ge 0$, we have $\alpha t \g... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $x \in A$ be [[Definition:Positive Element of C*-Algebra|positive]].
Let $\alpha \ge 0$.
Then $\alpha x$ is [[Definition:Positive Element of C*-Algebra|positive]]. | Let $\sigma_A$ denote the [[Definition:Spectrum (Spectral Theory)|spectrum in $A$]].
Since $x$ is [[Definition:Positive Element of C*-Algebra|positive]], we have:
:$\map {\sigma_A} x \subseteq \hointr 0 \infty$
From [[Spectral Mapping Theorem for Polynomials]], we have:
:$\map {\sigma_A} {\alpha x} = \set {\alpha t :... | Non-Negative Multiple of Positive Element of C*-Algebra is Positive | https://proofwiki.org/wiki/Non-Negative_Multiple_of_Positive_Element_of_C*-Algebra_is_Positive | https://proofwiki.org/wiki/Non-Negative_Multiple_of_Positive_Element_of_C*-Algebra_is_Positive | [
"Positive Elements of C*-Algebras",
"Non-Negative Multiple of Positive Element of C*-Algebra is Positive"
] | [
"Definition:C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Element of C*-Algebra"
] | [
"Definition:Spectrum (Spectral Theory)",
"Definition:Positive Element of C*-Algebra",
"Spectral Mapping Theorem for Polynomials",
"Definition:Involution on Algebra",
"Definition:Hermitian Element of *-Algebra",
"Definition:Real Number",
"Definition:Hermitian Element of *-Algebra",
"Definition:Positive... |
proofwiki-22121 | Characterization of Positive Element of Unital C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $a \in A$ be Hermitian.
Let $t \ge 0$.
Then if $\norm {a - t {\mathbf 1}_A} \le t$, $a$ is positive.
Conversely if $\norm a \le t$ and $a$ is positive, then $\norm {a - t {\mathbf 1}_A} \le t$. | Let $B$ to be $\text C^\ast$-algebra generated by $\set { {\mathbf 1}_A, x}$.
{{mistake|$x{{=}}a$}}
By C*-Algebra Generated by Commutative Self-Adjoint Set is Commutative, $B$ is commutative.
Further from Spectrum of Element of Unital C*-Subalgebra of Unital C*-Algebra we have $\map {\sigma_B} x = \map {\sigma_A} x$.
... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $a \in A$ be [[Definition:Hermitian Element of *-Algebra|Hermitian]].
Let $t \ge 0$.
Then if $\norm {a - t {\mathbf 1}_A} \le t$, $a$ is [[Definition:Positive Elemen... | Let $B$ to be [[Definition:Generated C*-Algebra|$\text C^\ast$-algebra generated]] by $\set { {\mathbf 1}_A, x}$.
{{mistake|$x{{=}}a$}}
By [[C*-Algebra Generated by Commutative Self-Adjoint Set is Commutative]], $B$ is [[Definition:Commutative Algebra (Abstract Algebra)|commutative]].
Further from [[Spectrum of Elem... | Characterization of Positive Element of Unital C*-Algebra | https://proofwiki.org/wiki/Characterization_of_Positive_Element_of_Unital_C*-Algebra | https://proofwiki.org/wiki/Characterization_of_Positive_Element_of_Unital_C*-Algebra | [
"Positive Elements of C*-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Hermitian Element of *-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Element of C*-Algebra"
] | [
"Definition:Generated C*-Algebra",
"C*-Algebra Generated by Commutative Self-Adjoint Set is Commutative",
"Definition:Commutative Algebra (Abstract Algebra)",
"Spectrum of Element of Unital C*-Subalgebra of Unital C*-Algebra",
"Definition:Spectrum of Banach Algebra",
"Gelfand-Naimark Theorem/Commutative C... |
proofwiki-22122 | Set of Positive Elements of C*-Algebra is Closed | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $A^+$ be the set of positive elements of $A$.
Then $A^+$ is closed. | Let $\struct {A_+, \ast, \norm {\, \cdot \,}_\ast}$ be the unitization of $A$.
Let ${\mathbf 1}_+$ be the identity element of $A_+$.
Let $\iota : A \to A_+$ be the mapping defined by:
:$\map \iota a = \tuple {a, 0}$
for each $a \in A$.
Let $\sequence {a_n}_{n \mathop \in \N}$ be a sequence in $A^+$ such that $a_n \t... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $A^+$ be the [[Definition:Set|set]] of [[Definition:Positive Element of C*-Algebra|positive elements]] of $A$.
Then $A^+$ is [[Definition:Closed Set|closed]]. | Let $\struct {A_+, \ast, \norm {\, \cdot \,}_\ast}$ be the [[Definition:Unitization of C*-Algebra|unitization]] of $A$.
Let ${\mathbf 1}_+$ be the [[Definition:Identity Element|identity element]] of $A_+$.
Let $\iota : A \to A_+$ be the [[Definition:Mapping|mapping]] defined by:
:$\map \iota a = \tuple {a, 0}$
for ... | Set of Positive Elements of C*-Algebra is Closed | https://proofwiki.org/wiki/Set_of_Positive_Elements_of_C*-Algebra_is_Closed | https://proofwiki.org/wiki/Set_of_Positive_Elements_of_C*-Algebra_is_Closed | [
"Positive Elements of C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Set",
"Definition:Positive Element of C*-Algebra",
"Definition:Closed Set"
] | [
"Definition:Unitization of C*-Algebra",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Mapping",
"Definition:Sequence",
"Convergence in Direct Product Norm",
"Definition:Convergent Sequence",
"Convergent Sequence in Normed Vector Space is Bounded",
"Element of C*-Algebra is P... |
proofwiki-22123 | Sum of Two Positive Elements of C*-Algebra is Positive | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $a, b$ be positive.
Then $a + b$ is positive. | From $(\text C^\ast 2)$ in the definition of an involution, $a + b$ is Hermitian.
First suppose that $A$ is unital.
From Characterization of Positive Element of Unital C*-Algebra we have:
:$\norm {a - \norm a {\mathbf 1}_A} \le \norm a$
and:
:$\norm {b - \norm b {\mathbf 1}_A} \le \norm b$
Hence we have:
{{begin-eqn}}... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $a, b$ be [[Definition:Positive Element of C*-Algebra|positive]].
Then $a + b$ is [[Definition:Positive Element of C*-Algebra|positive]]. | From $(\text C^\ast 2)$ in the definition of an [[Definition:Involution on Algebra|involution]], $a + b$ is [[Definition:Hermitian Element of *-Algebra|Hermitian]].
First suppose that $A$ is [[Definition:Unital Banach Algebra|unital]].
From [[Characterization of Positive Element of Unital C*-Algebra]] we have:
:$\nor... | Sum of Two Positive Elements of C*-Algebra is Positive | https://proofwiki.org/wiki/Sum_of_Two_Positive_Elements_of_C*-Algebra_is_Positive | https://proofwiki.org/wiki/Sum_of_Two_Positive_Elements_of_C*-Algebra_is_Positive | [
"Sum of Two Positive Elements of C*-Algebra is Positive",
"Positive Elements of C*-Algebras",
"Sum of Two Positive Elements of C*-Algebra is Positive"
] | [
"Definition:C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Element of C*-Algebra"
] | [
"Definition:Involution on Algebra",
"Definition:Hermitian Element of *-Algebra",
"Definition:Unital Banach Algebra",
"Characterization of Positive Element of Unital C*-Algebra",
"Characterization of Positive Element of Unital C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Unital B... |
proofwiki-22124 | Spectral Radius of Normal Element of C*-Algebra Equal to Norm/Corollary | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $r_A$ be the spectral radius in $A$.
Let $x \in A$ be normal such that:
:$\map {r_A} x = 0$
Then $x = {\mathbf 0}_A$. | From Spectral Radius of Normal Element of C*-Algebra Equal to Norm, we have:
:$\norm x = \map {r_A} x = 0$
Hence from {{NormAxiomVector|1}}, we have $x = {\mathbf 0}_A$.
{{qed}}
Category:Spectral Radius of Normal Element of C*-Algebra Equal to Norm
fp8t9u64etd6n80gvtopi9ny7qlzhbf | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $r_A$ be the [[Definition:Spectral Radius/Banach Algebra|spectral radius]] in $A$.
Let $x \in A$ be [[Definition:Normal Element of *-Algebra|normal]] such that:
:$\map {r_A} x = 0$
Then $x = {\mathbf 0}_A$. | From [[Spectral Radius of Normal Element of C*-Algebra Equal to Norm]], we have:
:$\norm x = \map {r_A} x = 0$
Hence from {{NormAxiomVector|1}}, we have $x = {\mathbf 0}_A$.
{{qed}}
[[Category:Spectral Radius of Normal Element of C*-Algebra Equal to Norm]]
fp8t9u64etd6n80gvtopi9ny7qlzhbf | Spectral Radius of Normal Element of C*-Algebra Equal to Norm/Corollary | https://proofwiki.org/wiki/Spectral_Radius_of_Normal_Element_of_C*-Algebra_Equal_to_Norm/Corollary | https://proofwiki.org/wiki/Spectral_Radius_of_Normal_Element_of_C*-Algebra_Equal_to_Norm/Corollary | [
"Spectral Radius of Normal Element of C*-Algebra Equal to Norm"
] | [
"Definition:C*-Algebra",
"Definition:Spectral Radius/Banach Algebra",
"Definition:Normal Element of *-Algebra"
] | [
"Spectral Radius of Normal Element of C*-Algebra Equal to Norm",
"Category:Spectral Radius of Normal Element of C*-Algebra Equal to Norm"
] |
proofwiki-22125 | Spectrum of Product of Elements of Banach Algebra | Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$.
Let $\sigma_A$ denote spectrum in $A$.
Let $x, y \in A$.
Then we have:
:$\map {\sigma_A} {x y} \setminus \set 0 = \map {\sigma_A} {y x} \setminus \set 0$ | First take $A$ to be unital.
We show that ${\mathbf 1}_A - x y$ is invertible {{iff}} ${\mathbf 1}_A - y x$ is invertible.
Swapping $x$ and $y$ it suffices to show that if ${\mathbf 1}_A - x y$ is invertible then ${\mathbf 1}_A - y x$ is invertible.
Suppose that ${\mathbf 1}_A - x y$ is invertible.
We have:
{{begin-eq... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Banach Algebra|Banach algebra]] over $\C$.
Let $\sigma_A$ denote [[Definition:Spectrum (Spectral Theory)|spectrum]] in $A$.
Let $x, y \in A$.
Then we have:
:$\map {\sigma_A} {x y} \setminus \set 0 = \map {\sigma_A} {y x} \setminus \set 0$ | First take $A$ to be [[Definition:Unital Algebra|unital]].
We show that ${\mathbf 1}_A - x y$ is [[Definition:Invertible Element|invertible]] {{iff}} ${\mathbf 1}_A - y x$ is [[Definition:Invertible Element|invertible]].
Swapping $x$ and $y$ it suffices to show that if ${\mathbf 1}_A - x y$ is [[Definition:Invertibl... | Spectrum of Product of Elements of Banach Algebra | https://proofwiki.org/wiki/Spectrum_of_Product_of_Elements_of_Banach_Algebra | https://proofwiki.org/wiki/Spectrum_of_Product_of_Elements_of_Banach_Algebra | [
"Spectrum (Spectral Theory)",
"Banach Algebras"
] | [
"Definition:Banach Algebra",
"Definition:Spectrum (Spectral Theory)"
] | [
"Definition:Unital Algebra",
"Definition:Invertible Element",
"Definition:Invertible Element",
"Definition:Invertible Element",
"Definition:Invertible Element",
"Definition:Invertible Element",
"Definition:Invertible Element",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Invertible E... |
proofwiki-22126 | Set of Positive Elements of C*-Algebra is Convex Cone | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $A^+$ be the set of positive elements of $A$.
Then $A^+$ is a convex cone. | From Non-Negative Multiple of Positive Element of C*-Algebra is Positive, if $x \in A^+$ and $\alpha \in \R_{\ge 0}$, we have $\alpha x \in A^+$.
Hence $A^+$ is a cone.
From Sum of Two Positive Elements of C*-Algebra is Positive, we have $x + y \in A^+$ for $x, y \in A^+$.
Hence $A^+$ is a convex cone.
{{qed}}
Categor... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $A^+$ be the [[Definition:Set|set]] of [[Definition:Positive Element of C*-Algebra|positive elements]] of $A$.
Then $A^+$ is a [[Definition:Convex Cone|convex cone]]. | From [[Non-Negative Multiple of Positive Element of C*-Algebra is Positive]], if $x \in A^+$ and $\alpha \in \R_{\ge 0}$, we have $\alpha x \in A^+$.
Hence $A^+$ is a [[Definition:Cone (Vector Space)|cone]].
From [[Sum of Two Positive Elements of C*-Algebra is Positive]], we have $x + y \in A^+$ for $x, y \in A^+$.
... | Set of Positive Elements of C*-Algebra is Convex Cone | https://proofwiki.org/wiki/Set_of_Positive_Elements_of_C*-Algebra_is_Convex_Cone | https://proofwiki.org/wiki/Set_of_Positive_Elements_of_C*-Algebra_is_Convex_Cone | [
"Positive Elements of C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Set",
"Definition:Positive Element of C*-Algebra",
"Definition:Convex Cone"
] | [
"Non-Negative Multiple of Positive Element of C*-Algebra is Positive",
"Definition:Cone (Vector Space)",
"Sum of Two Positive Elements of C*-Algebra is Positive",
"Definition:Convex Cone",
"Category:Positive Elements of C*-Algebras"
] |
proofwiki-22127 | Positive Part of Continuous Function is Continuous | Let $X$ be a topological space.
Let $f : X \to \R$ be continuous.
Let $f^+$ be the positive part of $f$.
Then $f^+$ is continuous. | From Constant Function is Continuous, we have that $\mathbf 0 : X \to \R$ defined by $\map {\mathbf 0} x = 0$ for each $x \in X$ is continuous.
From Maximum Rule for Continuous Real-Valued Functions, we have that the function $f \vee \mathbf 0 : X \to \R$ defined by:
:$\map {\paren {f \vee \mathbf 0} } x = \max \set {0... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $f : X \to \R$ be [[Definition:Continuous Real Function|continuous]].
Let $f^+$ be the [[Definition:Positive Part|positive part]] of $f$.
Then $f^+$ is [[Definition:Continuous Real Function|continuous]]. | From [[Constant Function is Continuous]], we have that $\mathbf 0 : X \to \R$ defined by $\map {\mathbf 0} x = 0$ for each $x \in X$ is [[Definition:Continuous Real Function|continuous]].
From [[Maximum Rule for Continuous Real-Valued Functions]], we have that the [[Definition:Real Function|function]] $f \vee \mathbf ... | Positive Part of Continuous Function is Continuous | https://proofwiki.org/wiki/Positive_Part_of_Continuous_Function_is_Continuous | https://proofwiki.org/wiki/Positive_Part_of_Continuous_Function_is_Continuous | [
"Positive Parts",
"Continuous Real Functions"
] | [
"Definition:Topological Space",
"Definition:Continuous Real Function",
"Definition:Positive Part",
"Definition:Continuous Real Function"
] | [
"Constant Function is Continuous",
"Definition:Continuous Real Function",
"Combination Theorem for Continuous Real-Valued Functions/Maximum Rule",
"Definition:Real Function",
"Definition:Continuous Real Function",
"Definition:Positive Part"
] |
proofwiki-22128 | Negative Part of Continuous Function is Continuous | Let $X$ be a topological space.
Let $f : X \to \R$ be continuous.
Let $f^-$ be the negative part of $f$.
Then $f^-$ is continuous. | From Constant Function is Continuous, we have that $\mathbf 0 : X \to \R$ defined by $\map {\mathbf 0} x = 0$ for each $x \in X$ is continuous.
From Minimum Rule for Continuous Real-Valued Functions, we have that the function $f \vee \mathbf 0 : X \to \R$ defined by:
:$\map {\paren {f \wedge \mathbf 0} } x = \min \set ... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $f : X \to \R$ be [[Definition:Continuous Real Function|continuous]].
Let $f^-$ be the [[Definition:Negative Part|negative part]] of $f$.
Then $f^-$ is [[Definition:Continuous Real Function|continuous]]. | From [[Constant Function is Continuous]], we have that $\mathbf 0 : X \to \R$ defined by $\map {\mathbf 0} x = 0$ for each $x \in X$ is [[Definition:Continuous Real Function|continuous]].
From [[Minimum Rule for Continuous Real-Valued Functions]], we have that the [[Definition:Real Function|function]] $f \vee \mathbf ... | Negative Part of Continuous Function is Continuous | https://proofwiki.org/wiki/Negative_Part_of_Continuous_Function_is_Continuous | https://proofwiki.org/wiki/Negative_Part_of_Continuous_Function_is_Continuous | [
"Negative Parts",
"Continuous Real Functions"
] | [
"Definition:Topological Space",
"Definition:Continuous Real Function",
"Definition:Negative Part",
"Definition:Continuous Real Function"
] | [
"Constant Function is Continuous",
"Definition:Continuous Real Function",
"Combination Theorem for Continuous Real-Valued Functions/Minimum Rule",
"Definition:Real Function",
"Definition:Continuous Real Function",
"Combination Theorem for Continuous Real-Valued Functions/Multiple Rule",
"Definition:Cont... |
proofwiki-22129 | Complex-Valued Function Dominated by Function Vanishing at Infinity also Vanishes at Infinity | Let $X$ be a topological space.
Let $g : X \to \C$ be a complex-valued function vanishing at infinity.
Let $f : X \to \C$ be such that $\cmod f \le \cmod g$.
Then $f$ vanishes at infinity. | Let $\epsilon > 0$.
Since $g$ vanishes at infinity, there exists a compact set $F \subseteq X$ such that:
:$\cmod {\map g x} < \epsilon$ for $x \in X \setminus F$.
We then have:
:$\cmod {\map f x} \le \cmod {\map g x} < \epsilon$ for $x \in X \setminus F$.
Since $\epsilon$ was arbitrary, we have that $f$ vanishes at i... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $g : X \to \C$ be a [[Definition:Complex-Valued Function Vanishing at Infinity|complex-valued function vanishing at infinity]].
Let $f : X \to \C$ be such that $\cmod f \le \cmod g$.
Then $f$ [[Definition:Complex-Valued Function Vanishing at Infi... | Let $\epsilon > 0$.
Since $g$ [[Definition:Complex-Valued Function Vanishing at Infinity|vanishes at infinity]], there exists a [[Definition:Compact Topological Space|compact set]] $F \subseteq X$ such that:
:$\cmod {\map g x} < \epsilon$ for $x \in X \setminus F$.
We then have:
:$\cmod {\map f x} \le \cmod {\map g ... | Complex-Valued Function Dominated by Function Vanishing at Infinity also Vanishes at Infinity | https://proofwiki.org/wiki/Complex-Valued_Function_Dominated_by_Function_Vanishing_at_Infinity_also_Vanishes_at_Infinity | https://proofwiki.org/wiki/Complex-Valued_Function_Dominated_by_Function_Vanishing_at_Infinity_also_Vanishes_at_Infinity | [
"Complex-Valued Functions Vanishing at Infinity"
] | [
"Definition:Topological Space",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Complex-Valued Function Vanishing at Infinity"
] | [
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Compact Topological Space",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Category:Complex-Valued Functions Vanishing at Infinity"
] |
proofwiki-22130 | Positive Part of Real-Valued Function Vanishing at Infinity Vanishes at Infinity | Let $X$ be a topological space.
Let $f : X \to \R$ be a real-valued function vanishing at infinity.
Let $f^+$ be the positive part of $f$.
Then $f^+$ vanishes at infinity. | From Positive Part of Function Bounded above by Absolute Value, we have:
:$0 \le f^+ \le \cmod f$
From Complex-Valued Function Dominated by Function Vanishing at Infinity also Vanishes at Infinity:
:$f^+$ vanishes at infinity.
{{qed}}
Category:Positive Parts
Category:Complex-Valued Functions Vanishing at Infinity
gbm9b... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $f : X \to \R$ be a [[Definition:Complex-Valued Function Vanishing at Infinity|real-valued function vanishing at infinity]].
Let $f^+$ be the [[Definition:Positive Part|positive part]] of $f$.
Then $f^+$ [[Definition:Complex-Valued Function Vani... | From [[Positive Part of Function Bounded above by Absolute Value]], we have:
:$0 \le f^+ \le \cmod f$
From [[Complex-Valued Function Dominated by Function Vanishing at Infinity also Vanishes at Infinity]]:
:$f^+$ [[Definition:Complex-Valued Function Vanishing at Infinity|vanishes at infinity]].
{{qed}}
[[Category:Pos... | Positive Part of Real-Valued Function Vanishing at Infinity Vanishes at Infinity | https://proofwiki.org/wiki/Positive_Part_of_Real-Valued_Function_Vanishing_at_Infinity_Vanishes_at_Infinity | https://proofwiki.org/wiki/Positive_Part_of_Real-Valued_Function_Vanishing_at_Infinity_Vanishes_at_Infinity | [
"Positive Parts",
"Complex-Valued Functions Vanishing at Infinity"
] | [
"Definition:Topological Space",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Positive Part",
"Definition:Complex-Valued Function Vanishing at Infinity"
] | [
"Positive Part of Function Bounded above by Absolute Value",
"Complex-Valued Function Dominated by Function Vanishing at Infinity also Vanishes at Infinity",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Category:Positive Parts",
"Category:Complex-Valued Functions Vanishing at Infinity"
] |
proofwiki-22131 | Positive Part of Function Bounded above by Absolute Value | Let $X$ be a set.
Let $f : X \to \C$ be a function.
Let $f^+$ be the positive part of $f$.
Then $f^+ \le \cmod f$. | By definition, we have $f^+ \ge 0$.
Let $f^-$ be the negative part of $f$.
By definition, we also have $f^- \ge 0$.
Hence $f^+ \le f^+ + f^-$.
By Sum of Positive and Negative Parts, we have $f^+ + f^- = \cmod f$.
Hence $0 \le f^+ \le \cmod f$.
{{qed}}
Category:Positive Parts
e47s1edq6a1qzltnrxefsn2bsag0nc1 | Let $X$ be a [[Definition:Set|set]].
Let $f : X \to \C$ be a [[Definition:Function|function]].
Let $f^+$ be the [[Definition:Positive Part|positive part]] of $f$.
Then $f^+ \le \cmod f$. | By definition, we have $f^+ \ge 0$.
Let $f^-$ be the [[Definition:Negative Part|negative part]] of $f$.
By definition, we also have $f^- \ge 0$.
Hence $f^+ \le f^+ + f^-$.
By [[Sum of Positive and Negative Parts]], we have $f^+ + f^- = \cmod f$.
Hence $0 \le f^+ \le \cmod f$.
{{qed}}
[[Category:Positive Parts]... | Positive Part of Function Bounded above by Absolute Value | https://proofwiki.org/wiki/Positive_Part_of_Function_Bounded_above_by_Absolute_Value | https://proofwiki.org/wiki/Positive_Part_of_Function_Bounded_above_by_Absolute_Value | [
"Positive Parts"
] | [
"Definition:Set",
"Definition:Function",
"Definition:Positive Part"
] | [
"Definition:Negative Part",
"Sum of Positive and Negative Parts",
"Category:Positive Parts"
] |
proofwiki-22132 | Negative Part of Function Bounded above by Absolute Value | Let $X$ be a set.
Let $f : X \to \C$ be a function.
Let $f^-$ be the negative part of $f$.
Then $f^- \le \cmod f$. | By definition, we have $f^- \ge 0$.
Let $f^+$ be the positive part of $f$.
By definition, we also have $f^+ \ge 0$.
Hence $f^- \le f^+ + f^-$.
By Sum of Positive and Negative Parts, we have $f^+ + f^- = \cmod f$.
Hence $0 \le f^- \le \cmod f$.
{{qed}}
Category:Negative Parts
fjk0mt1cipzitewb2qrscsiz8th52v5 | Let $X$ be a [[Definition:Set|set]].
Let $f : X \to \C$ be a [[Definition:Function|function]].
Let $f^-$ be the [[Definition:Negative Part|negative part]] of $f$.
Then $f^- \le \cmod f$. | By definition, we have $f^- \ge 0$.
Let $f^+$ be the [[Definition:Positive Part|positive part]] of $f$.
By definition, we also have $f^+ \ge 0$.
Hence $f^- \le f^+ + f^-$.
By [[Sum of Positive and Negative Parts]], we have $f^+ + f^- = \cmod f$.
Hence $0 \le f^- \le \cmod f$.
{{qed}}
[[Category:Negative Parts]... | Negative Part of Function Bounded above by Absolute Value | https://proofwiki.org/wiki/Negative_Part_of_Function_Bounded_above_by_Absolute_Value | https://proofwiki.org/wiki/Negative_Part_of_Function_Bounded_above_by_Absolute_Value | [
"Negative Parts"
] | [
"Definition:Set",
"Definition:Function",
"Definition:Negative Part"
] | [
"Definition:Positive Part",
"Sum of Positive and Negative Parts",
"Category:Negative Parts"
] |
proofwiki-22133 | Negative Part of Real-Valued Function Vanishing at Infinity Vanishes at Infinity | Let $X$ be a topological space.
Let $f : X \to \R$ be a real-valued function vanishing at infinity.
Let $f^-$ be the negative part of $f$.
Then $f^-$ vanishes at infinity. | From Negative Part of Function Bounded above by Absolute Value, we have:
:$0 \le f^- \le \cmod f$
From Complex-Valued Function Dominated by Function Vanishing at Infinity also Vanishes at Infinity:
:$f^-$ vanishes at infinity.
{{qed}}
Category:Negative Parts
Category:Complex-Valued Functions Vanishing at Infinity
8o73q... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $f : X \to \R$ be a [[Definition:Complex-Valued Function Vanishing at Infinity|real-valued function vanishing at infinity]].
Let $f^-$ be the [[Definition:Negative Part|negative part]] of $f$.
Then $f^-$ [[Definition:Complex-Valued Function Vani... | From [[Negative Part of Function Bounded above by Absolute Value]], we have:
:$0 \le f^- \le \cmod f$
From [[Complex-Valued Function Dominated by Function Vanishing at Infinity also Vanishes at Infinity]]:
:$f^-$ [[Definition:Complex-Valued Function Vanishing at Infinity|vanishes at infinity]].
{{qed}}
[[Category:Neg... | Negative Part of Real-Valued Function Vanishing at Infinity Vanishes at Infinity | https://proofwiki.org/wiki/Negative_Part_of_Real-Valued_Function_Vanishing_at_Infinity_Vanishes_at_Infinity | https://proofwiki.org/wiki/Negative_Part_of_Real-Valued_Function_Vanishing_at_Infinity_Vanishes_at_Infinity | [
"Negative Parts",
"Complex-Valued Functions Vanishing at Infinity"
] | [
"Definition:Topological Space",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Negative Part",
"Definition:Complex-Valued Function Vanishing at Infinity"
] | [
"Negative Part of Function Bounded above by Absolute Value",
"Complex-Valued Function Dominated by Function Vanishing at Infinity also Vanishes at Infinity",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Category:Negative Parts",
"Category:Complex-Valued Functions Vanishing at Infinity"
] |
proofwiki-22134 | Product of Element of C*-Algebra with its Star is Positive | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $a \in A$.
Then $a^\ast a$ is positive. | We first show that if $-a^\ast a$ is positive then $a = {\mathbf 0}_A$.
From Product of Element in *-Star Algebra with its Star is Hermitian, $a^\ast a$ and $a a^\ast$ are Hermitian.
Suppose that $-a^\ast a$ is positive.
From Spectrum of Product of Elements of Banach Algebra, we have:
:$\map {\sigma_A} {-a^\ast a} \s... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $a \in A$.
Then $a^\ast a$ is [[Definition:Positive Element of C*-Algebra|positive]]. | We first show that if $-a^\ast a$ is [[Definition:Positive Element of C*-Algebra|positive]] then $a = {\mathbf 0}_A$.
From [[Product of Element in *-Star Algebra with its Star is Hermitian]], $a^\ast a$ and $a a^\ast$ are [[Definition:Hermitian Element of *-Algebra|Hermitian]].
Suppose that $-a^\ast a$ is [[Definit... | Product of Element of C*-Algebra with its Star is Positive | https://proofwiki.org/wiki/Product_of_Element_of_C*-Algebra_with_its_Star_is_Positive | https://proofwiki.org/wiki/Product_of_Element_of_C*-Algebra_with_its_Star_is_Positive | [
"Positive Elements of C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Positive Element of C*-Algebra"
] | [
"Definition:Positive Element of C*-Algebra",
"Product of Element in *-Star Algebra with its Star is Hermitian",
"Definition:Hermitian Element of *-Algebra",
"Definition:Positive Element of C*-Algebra",
"Spectrum of Product of Elements of Banach Algebra",
"Definition:Spectrum (Spectral Theory)",
"Definit... |
proofwiki-22135 | Product of Positive and Negative Parts is Zero | Let $X$ be a set.
Let $f : X \to \R$ be a real-valued function.
Let $f^+$ and $f^-$ be the positive part and negative part of $f$ respectively.
Then $f^+ f^- = 0$. | Let $x \in X$.
If $\map f x = 0$, then we have $\map {f^+} x = \map {f^-} x = 0$.
Hence $\map {f^+} x = \map {f^-} x = 0$ in this case.
If $\map f x > 0$, then $\map {f^+} x = \map f x$ and $\map {f^-} x = 0$.
So $\map {f^+} x \map {f^-} x = 0$ in this case also.
Finally, if $\map f x < 0$, then $\map {f^+} x = 0$ and... | Let $X$ be a [[Definition:Set|set]].
Let $f : X \to \R$ be a [[Definition:Real-Valued Function|real-valued function]].
Let $f^+$ and $f^-$ be the [[Definition:Positive Part|positive part]] and [[Definition:Negative Part|negative part]] of $f$ respectively.
Then $f^+ f^- = 0$. | Let $x \in X$.
If $\map f x = 0$, then we have $\map {f^+} x = \map {f^-} x = 0$.
Hence $\map {f^+} x = \map {f^-} x = 0$ in this case.
If $\map f x > 0$, then $\map {f^+} x = \map f x$ and $\map {f^-} x = 0$.
So $\map {f^+} x \map {f^-} x = 0$ in this case also.
Finally, if $\map f x < 0$, then $\map {f^+} x = 0... | Product of Positive and Negative Parts is Zero | https://proofwiki.org/wiki/Product_of_Positive_and_Negative_Parts_is_Zero | https://proofwiki.org/wiki/Product_of_Positive_and_Negative_Parts_is_Zero | [
"Positive Parts",
"Negative Parts"
] | [
"Definition:Set",
"Definition:Real-Valued Function",
"Definition:Positive Part",
"Definition:Negative Part"
] | [
"Category:Positive Parts",
"Category:Negative Parts"
] |
proofwiki-22136 | Set of Positive Elements of C*-Algebra is Set of Products of Element with its Star | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $a \in A$.
Let $A^+$ be the set of positive elements of $A$.
We have:
:$A^+ = \set {a^\ast a : a \in A}$ | From Product of Element of C*-Algebra with its Star is Positive, we have:
:$\set {a^\ast a : a \in A} \subseteq A^+$
Conversely suppose that $x \in A^+$.
From Existence and Uniqueness of Positive Nth Root of Positive Element of C*-Algebra, there exists $y \in A^+$ such that $x = y^2$.
Since $y$ is Hermitian, we have $x... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $a \in A$.
Let $A^+$ be the [[Definition:Set|set]] of [[Definition:Positive Element of C*-Algebra|positive elements]] of $A$.
We have:
:$A^+ = \set {a^\ast a : a \in A}$ | From [[Product of Element of C*-Algebra with its Star is Positive]], we have:
:$\set {a^\ast a : a \in A} \subseteq A^+$
Conversely suppose that $x \in A^+$.
From [[Existence and Uniqueness of Positive Nth Root of Positive Element of C*-Algebra]], there exists $y \in A^+$ such that $x = y^2$.
Since $y$ is [[Definiti... | Set of Positive Elements of C*-Algebra is Set of Products of Element with its Star | https://proofwiki.org/wiki/Set_of_Positive_Elements_of_C*-Algebra_is_Set_of_Products_of_Element_with_its_Star | https://proofwiki.org/wiki/Set_of_Positive_Elements_of_C*-Algebra_is_Set_of_Products_of_Element_with_its_Star | [
"Positive Elements of C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Set",
"Definition:Positive Element of C*-Algebra"
] | [
"Product of Element of C*-Algebra with its Star is Positive",
"Existence and Uniqueness of Positive Nth Root of Positive Element of C*-Algebra",
"Definition:Hermitian Element of *-Algebra"
] |
proofwiki-22137 | Conjugation in C*-Algebra preserves Positivity | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $a \in A$ be positive.
Let $c \in A$.
Then $c^\ast a c$ is positive. | Since $a$ is positive, there exists a positive $b \in A$ such that $a = b^2$ from Existence and Uniqueness of Positive Nth Root of Positive Element of C*-Algebra.
We then have:
:$\paren {b c}^\ast \paren {b c} = c^\ast b^2 c = c^\ast a c$
From Product of Element of C*-Algebra with its Star is Positive, we have that $\p... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $a \in A$ be [[Definition:Positive Element of C*-Algebra|positive]].
Let $c \in A$.
Then $c^\ast a c$ is [[Definition:Positive Element of C*-Algebra|positive]]. | Since $a$ is [[Definition:Positive Element of C*-Algebra|positive]], there exists a [[Definition:Positive Element of C*-Algebra|positive]] $b \in A$ such that $a = b^2$ from [[Existence and Uniqueness of Positive Nth Root of Positive Element of C*-Algebra]].
We then have:
:$\paren {b c}^\ast \paren {b c} = c^\ast b^2 ... | Conjugation in C*-Algebra preserves Positivity | https://proofwiki.org/wiki/Conjugation_in_C*-Algebra_preserves_Positivity | https://proofwiki.org/wiki/Conjugation_in_C*-Algebra_preserves_Positivity | [
"Positive Elements of C*-Algebras",
"Conjugation in C*-Algebra preserves Positivity"
] | [
"Definition:C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Element of C*-Algebra"
] | [
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Existence and Uniqueness of Positive Nth Root of Positive Element of C*-Algebra",
"Product of Element of C*-Algebra with its Star is Positive",
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Eleme... |
proofwiki-22138 | Conjugation in C*-Algebra preserves Positivity/Corollary | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $\le$ be the canonical preordering on $A$.
Let $a, b \in A$ be such that $a \le b$.
Let $c \in A$.
Then $c^\ast a c \le c^\ast b c$. | Since $a \le b$, we have that $b - a$ is positive.
Hence $c^\ast \paren {b - a} c$ is positive by Conjugation in C*-Algebra preserves Positivity.
Hence $c^\ast b c - c^\ast a c$ is positive.
By the definition of $\le$, we have $c^\ast a c \le c^\ast b c$.
{{qed}} | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\le$ be the [[Definition:Canonical Preordering of C*-Algebra|canonical preordering]] on $A$.
Let $a, b \in A$ be such that $a \le b$.
Let $c \in A$.
Then $c^\ast a c \le c^\ast b c$. | Since $a \le b$, we have that $b - a$ is [[Definition:Positive Element of C*-Algebra|positive]].
Hence $c^\ast \paren {b - a} c$ is [[Definition:Positive Element of C*-Algebra|positive]] by [[Conjugation in C*-Algebra preserves Positivity]].
Hence $c^\ast b c - c^\ast a c$ is [[Definition:Positive Element of C*-Algeb... | Conjugation in C*-Algebra preserves Positivity/Corollary | https://proofwiki.org/wiki/Conjugation_in_C*-Algebra_preserves_Positivity/Corollary | https://proofwiki.org/wiki/Conjugation_in_C*-Algebra_preserves_Positivity/Corollary | [
"Conjugation in C*-Algebra preserves Positivity"
] | [
"Definition:C*-Algebra",
"Definition:Canonical Preordering of C*-Algebra"
] | [
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Conjugation in C*-Algebra preserves Positivity",
"Definition:Positive Element of C*-Algebra"
] |
proofwiki-22139 | Non-Negative Multiple of Positive Element of C*-Algebra is Positive/Corollary | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $\le$ be the canonical ordering of $A$.
Let $x \in A$ be positive.
Let $\alpha, \beta \in \R$ have $\alpha \le \beta$.
Then we have:
:$\alpha x \le \beta x$ | Since $\alpha \le \beta$, we have that $\beta - \alpha \ge 0$ and hence that $\paren {\beta - \alpha} x$ is positive.
Hence $\beta x - \alpha x$ is positive.
Hence we have $\alpha x \le \beta x$ by the definition of $\le$.
{{qed}}
Category:Non-Negative Multiple of Positive Element of C*-Algebra is Positive
e5sq611dhpb... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\le$ be the [[Definition:Canonical Preordering of C*-Algebra|canonical ordering]] of $A$.
Let $x \in A$ be [[Definition:Positive Element of C*-Algebra|positive]].
Let $\alpha, \beta \in \R$ have $\alpha \le \be... | Since $\alpha \le \beta$, we have that $\beta - \alpha \ge 0$ and hence that $\paren {\beta - \alpha} x$ is [[Definition:Positive Element of C*-Algebra|positive]].
Hence $\beta x - \alpha x$ is [[Definition:Positive Element of C*-Algebra|positive]].
Hence we have $\alpha x \le \beta x$ by the definition of $\le$.
{{... | Non-Negative Multiple of Positive Element of C*-Algebra is Positive/Corollary | https://proofwiki.org/wiki/Non-Negative_Multiple_of_Positive_Element_of_C*-Algebra_is_Positive/Corollary | https://proofwiki.org/wiki/Non-Negative_Multiple_of_Positive_Element_of_C*-Algebra_is_Positive/Corollary | [
"Non-Negative Multiple of Positive Element of C*-Algebra is Positive"
] | [
"Definition:C*-Algebra",
"Definition:Canonical Preordering of C*-Algebra",
"Definition:Positive Element of C*-Algebra"
] | [
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Category:Non-Negative Multiple of Positive Element of C*-Algebra is Positive"
] |
proofwiki-22140 | Identity Element is Order Unit on Set of Hermitian Elements of Unital C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $\le$ be the canonical preordering on $A$.
Let $A_{\mathbf {SA}}$ be the set of Hermitian elements of $A$.
Then ${\mathbf 1}_A$ is an order unit for $\tuple {A_{\mathbf{SA}}, \le}$. | Fix $a \in A$ Hermitian.
Then from Bounds on Hermitian Element of Unital C*-Algebra in terms of Bounds on Spectrum:
:$-\norm a {\mathbf 1}_A \le_A a \le \norm a {\mathbf 1}_A$
Take $N \in \N$ such that $N > \norm a$.
From Non-Negative Multiple of Positive Element of C*-Algebra is Positive: Corollary, we have:
:$-N {\m... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\le$ be the [[Definition:Canonical Preordering of C*-Algebra|canonical preordering]] on $A$.
Let $A_{\mathbf {SA}}$ be the set of [[Definition:Hermitian Element of *-Al... | Fix $a \in A$ [[Definition:Hermitian Element of *-Algebra|Hermitian]].
Then from [[Bounds on Hermitian Element of Unital C*-Algebra in terms of Bounds on Spectrum]]:
:$-\norm a {\mathbf 1}_A \le_A a \le \norm a {\mathbf 1}_A$
Take $N \in \N$ such that $N > \norm a$.
From [[Non-Negative Multiple of Positive Element ... | Identity Element is Order Unit on Set of Hermitian Elements of Unital C*-Algebra | https://proofwiki.org/wiki/Identity_Element_is_Order_Unit_on_Set_of_Hermitian_Elements_of_Unital_C*-Algebra | https://proofwiki.org/wiki/Identity_Element_is_Order_Unit_on_Set_of_Hermitian_Elements_of_Unital_C*-Algebra | [
"C*-Algebras",
"Preordered Vector Spaces"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Canonical Preordering of C*-Algebra",
"Definition:Hermitian Element of *-Algebra",
"Definition:Order Unit"
] | [
"Definition:Hermitian Element of *-Algebra",
"Bounds on Hermitian Element of Unital C*-Algebra in terms of Bounds on Spectrum",
"Non-Negative Multiple of Positive Element of C*-Algebra is Positive/Corollary",
"Definition:Preordering",
"Definition:Hermitian Element of *-Algebra",
"Definition:Order Unit",
... |
proofwiki-22141 | Norm Preserves Ordering on Positive Elements of C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $\le_A$ be the canonical preordering on $A$.
Let $a, b \in A$ be such that:
:${\mathbf 0}_A \le_A a \le_A b$
Then:
:$\norm a \le \norm b$ | First take $A$ unital.
From Identity Element is Order Unit on Set of Hermitian Elements of Unital C*-Algebra, we obtain:
:$b \le_A \norm b {\mathbf 1}_A$
Since $\le_A$ is a preordering and $a \le_A b$, we have:
:${\mathbf 0}_A \le_A a \le_A \norm b {\mathbf 1}_A$
Let $B$ be the $\text C^\ast$-algebra generated by $\set... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\le_A$ be the [[Definition:Canonical Preordering of C*-Algebra|canonical preordering]] on $A$.
Let $a, b \in A$ be such that:
:${\mathbf 0}_A \le_A a \le_A b$
Then:
:$\norm a \le \norm b$ | First take $A$ [[Definition:Unital Banach Algebra|unital]].
From [[Identity Element is Order Unit on Set of Hermitian Elements of Unital C*-Algebra]], we obtain:
:$b \le_A \norm b {\mathbf 1}_A$
Since $\le_A$ is a [[Definition:Preordering|preordering]] and $a \le_A b$, we have:
:${\mathbf 0}_A \le_A a \le_A \norm b {... | Norm Preserves Ordering on Positive Elements of C*-Algebra | https://proofwiki.org/wiki/Norm_Preserves_Ordering_on_Positive_Elements_of_C*-Algebra | https://proofwiki.org/wiki/Norm_Preserves_Ordering_on_Positive_Elements_of_C*-Algebra | [
"Canonical Preorderings on C*-Algebras",
"Canonical Preorderings of C*-Algebras",
"Canonical Preorderings of C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Canonical Preordering of C*-Algebra"
] | [
"Definition:Unital Banach Algebra",
"Identity Element is Order Unit on Set of Hermitian Elements of Unital C*-Algebra",
"Definition:Preordering",
"Definition:Generated C*-Algebra",
"Subalgebra Generated by Commuting Elements is Commutative",
"Definition:Commutative Algebra (Abstract Algebra)",
"Definiti... |
proofwiki-22142 | Normal Distribution is Pearson Distribution | The normal distribution is an example of a Pearson distribution. | Recall the definition of the Pearson distribution:
{{:Definition:Pearson Distribution}}
Setting $a = -1$, $b = c = d = 0$ gives:
:$\map {f'} x = -x \map f x$
{{ProofWanted|It remains to be shown that the solution to the above is indeed a normal distribution.}} | The [[Definition:Normal Distribution|normal distribution]] is an example of a [[Definition:Pearson Distribution|Pearson distribution]]. | Recall the definition of the [[Definition:Pearson Distribution|Pearson distribution]]:
{{:Definition:Pearson Distribution}}
Setting $a = -1$, $b = c = d = 0$ gives:
:$\map {f'} x = -x \map f x$
{{ProofWanted|It remains to be shown that the solution to the above is indeed a [[Definition:Normal Distribution|normal dis... | Normal Distribution is Pearson Distribution | https://proofwiki.org/wiki/Normal_Distribution_is_Pearson_Distribution | https://proofwiki.org/wiki/Normal_Distribution_is_Pearson_Distribution | [
"Normal Distribution",
"Pearson Distributions"
] | [
"Definition:Normal Distribution",
"Definition:Pearson Distribution"
] | [
"Definition:Pearson Distribution",
"Definition:Normal Distribution"
] |
proofwiki-22143 | Bounds on Hermitian Element of Unital C*-Algebra in terms of Bounds on Spectrum | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $\le_A$ be the canonical preordering on $A$.
Let $a \in A$ be Hermitian.
Then we have $\map {\sigma_A} x \subseteq \closedint \alpha \beta$ {{iff}}:
:$\alpha {\mathbf 1}_A \le_A a \le_A \beta {\mathbf 1}_A$
In particular:
:$-\norm a {... | === Necessary Condition ===
Since $a$ is Hermitian, and ${\mathbf 1}_A$ is Hermitian from Identity Element in Unital *-Algebra is Hermitian, we have:
:$a - \lambda {\mathbf 1}_A$ is Hermitian for each $\lambda \in \R$.
From Spectral Mapping Theorem for Polynomials, we have:
:$\map {\sigma_A} {a - \alpha {\mathbf 1}_A} ... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\le_A$ be the [[Definition:Canonical Preordering of C*-Algebra|canonical preordering]] on $A$.
Let $a \in A$ be [[Definition:Hermitian Element of *-Algebra|Hermitian]].... | === Necessary Condition ===
Since $a$ is [[Definition:Hermitian Element of *-Algebra|Hermitian]], and ${\mathbf 1}_A$ is [[Definition:Hermitian Element of *-Algebra|Hermitian]] from [[Identity Element in Unital *-Algebra is Hermitian]], we have:
:$a - \lambda {\mathbf 1}_A$ is [[Definition:Hermitian Element of *-Algeb... | Bounds on Hermitian Element of Unital C*-Algebra in terms of Bounds on Spectrum | https://proofwiki.org/wiki/Bounds_on_Hermitian_Element_of_Unital_C*-Algebra_in_terms_of_Bounds_on_Spectrum | https://proofwiki.org/wiki/Bounds_on_Hermitian_Element_of_Unital_C*-Algebra_in_terms_of_Bounds_on_Spectrum | [
"Canonical Preorderings of C*-Algebras",
"Hermitian Elements of *-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Canonical Preordering of C*-Algebra",
"Definition:Hermitian Element of *-Algebra"
] | [
"Definition:Hermitian Element of *-Algebra",
"Definition:Hermitian Element of *-Algebra",
"Identity Element in Unital *-Algebra is Hermitian",
"Definition:Hermitian Element of *-Algebra",
"Spectral Mapping Theorem for Polynomials",
"Definition:Positive Element of C*-Algebra",
"Spectral Mapping Theorem f... |
proofwiki-22144 | Image of Positive Element of C*-Algebra under *-Algebra Homomorphism is Positive | Let $\struct {A, \ast, \norm {\, \cdot \,}_A}$ and $\struct {B, \square, \norm {\, \cdot \,}_B}$ be $\text C^\ast$-algebras.
Let $\phi : A \to B$ be a $\ast$-algebra homomorphism.
Let $a \in A$ be positive.
Then $\map \phi a$ is positive. | We first show that $\map \phi a \in B$ is Hermitian.
Since $\phi$ is a $\ast$-algebra isomorphism, we have:
:$\map \phi a^\square = \map \phi {a^\ast} = \map \phi a$
Hence $\map \phi a$ is Hermitian.
From Spectrum of Image of Element of Unital Algebra under Unital Algebra Homomorphism: Corollary, we have:
:$\map {\sigm... | Let $\struct {A, \ast, \norm {\, \cdot \,}_A}$ and $\struct {B, \square, \norm {\, \cdot \,}_B}$ be [[Definition:C*-Algebra|$\text C^\ast$-algebras]].
Let $\phi : A \to B$ be a [[Definition:*-Algebra Homomorphism|$\ast$-algebra homomorphism]].
Let $a \in A$ be [[Definition:Positive Element of C*-Algebra|positive]].
... | We first show that $\map \phi a \in B$ is [[Definition:Hermitian Element of *-Algebra|Hermitian]].
Since $\phi$ is a [[Definition:*-Algebra Isomorphism|$\ast$-algebra isomorphism]], we have:
:$\map \phi a^\square = \map \phi {a^\ast} = \map \phi a$
Hence $\map \phi a$ is [[Definition:Hermitian Element of *-Algebra|He... | Image of Positive Element of C*-Algebra under *-Algebra Homomorphism is Positive | https://proofwiki.org/wiki/Image_of_Positive_Element_of_C*-Algebra_under_*-Algebra_Homomorphism_is_Positive | https://proofwiki.org/wiki/Image_of_Positive_Element_of_C*-Algebra_under_*-Algebra_Homomorphism_is_Positive | [
"Positive Elements of C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:*-Algebra Homomorphism",
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Element of C*-Algebra"
] | [
"Definition:Hermitian Element of *-Algebra",
"Definition:*-Algebra Isomorphism",
"Definition:Hermitian Element of *-Algebra",
"Spectrum of Image of Element of Unital Algebra under Unital Algebra Homomorphism/Corollary",
"Definition:Positive Element of C*-Algebra",
"Category:Positive Elements of C*-Algebra... |
proofwiki-22145 | Spectrum of Image of Element of Unital Algebra under Unital Algebra Homomorphism/Corollary | Let $A, B$ be algebras over $\C$.
Let $\phi : A \to B$ be an algebra homomorphism.
Let $x \in A$.
Then:
:$\map {\sigma_B} {\map \phi x} \cup \set 0 \subseteq \map {\sigma_A} x \cup \set 0$
where $\sigma_A$ and $\sigma_B$ denote spectra in $A$ and $B$ respectively. | Let $A_+$ and $B_+$ be the unitizations of $A$ and $B$ respectively.
Define $\psi : A_+ \to B_+$ by:
:$\map \psi {a, \lambda} = \tuple {\map \phi a, \lambda}$
for each $\tuple {a, \lambda} \in A_+$.
From Induced Algebra Homomorphism on Unitization, $\psi$ is a unital algebra homomorphism.
Hence we have:
:$\map {\sigma... | Let $A, B$ be [[Definition:Algebra over Field|algebras]] over $\C$.
Let $\phi : A \to B$ be an [[Definition:Algebra Homomorphism|algebra homomorphism]].
Let $x \in A$.
Then:
:$\map {\sigma_B} {\map \phi x} \cup \set 0 \subseteq \map {\sigma_A} x \cup \set 0$
where $\sigma_A$ and $\sigma_B$ denote [[Definition:Spect... | Let $A_+$ and $B_+$ be the [[Definition:Unitization of Algebra over Field|unitizations]] of $A$ and $B$ respectively.
Define $\psi : A_+ \to B_+$ by:
:$\map \psi {a, \lambda} = \tuple {\map \phi a, \lambda}$
for each $\tuple {a, \lambda} \in A_+$.
From [[Induced Algebra Homomorphism on Unitization]], $\psi$ is a [[D... | Spectrum of Image of Element of Unital Algebra under Unital Algebra Homomorphism/Corollary | https://proofwiki.org/wiki/Spectrum_of_Image_of_Element_of_Unital_Algebra_under_Unital_Algebra_Homomorphism/Corollary | https://proofwiki.org/wiki/Spectrum_of_Image_of_Element_of_Unital_Algebra_under_Unital_Algebra_Homomorphism/Corollary | [
"Spectrum of Image of Element of Unital Algebra under Unital Algebra Homomorphism"
] | [
"Definition:Algebra over Field",
"Definition:Algebra Homomorphism",
"Definition:Spectrum (Spectral Theory)/Unital Algebra"
] | [
"Definition:Unitization of Algebra over Field",
"Induced Algebra Homomorphism on Unitization",
"Definition:Unital Algebra Homomorphism",
"Spectrum of Element in Unitization of Unital Algebra",
"Category:Spectrum of Image of Element of Unital Algebra under Unital Algebra Homomorphism"
] |
proofwiki-22146 | Solution to Pell's Equation | Recall Pell's equation:
{{:Definition:Pell's Equation}}
Let the continued fraction of $\sqrt n$ have a cycle whose length is $s$:
:$\sqrt n = \sqbrk {a_1 \sequence {a_2, a_3, \ldots, a_{s + 1} } }$
Let $a_n = \dfrac {p_n} {q_n}$ be a convergent of $\sqrt n$.
Then:
:${p_{r s} }^2 - n {q_{r s} }^2 = \paren {-1}^{r s}$ fo... | First note that if $x = p, y = q$ is a positive solution of $x^2 - n y^2 = 1$ then $\dfrac p q$ is a convergent of $\sqrt n$.
The continued fraction of $\sqrt n$ is periodic from Continued Fraction Expansion of Irrational Square Root and of the form:
:$\sqbrk {a \sequence {b_1, b_2, \ldots, b_{m - 1}, b_m, b_{m - 1}, \... | Recall [[Definition:Pell's Equation|Pell's equation]]:
{{:Definition:Pell's Equation}}
Let the [[Definition:Continued Fraction|continued fraction]] of $\sqrt n$ have a [[Definition:Cycle of Periodic Continued Fraction|cycle]] whose [[Definition:Cycle Length of Periodic Continued Fraction|length]] is $s$:
:$\sqrt n =... | First note that if $x = p, y = q$ is a [[Definition:Positive Integer|positive]] solution of $x^2 - n y^2 = 1$ then [[Solution of Pell's Equation is a Convergent|$\dfrac p q$ is a convergent of $\sqrt n$]].
The [[Definition:Continued Fraction|continued fraction]] of $\sqrt n$ is [[Definition:Periodic Continued Fractio... | Solution to Pell's Equation | https://proofwiki.org/wiki/Solution_to_Pell's_Equation | https://proofwiki.org/wiki/Solution_to_Pell's_Equation | [
"Pell's Equation"
] | [
"Definition:Pell's Equation",
"Definition:Continued Fraction",
"Definition:Periodic Continued Fraction/Cycle",
"Definition:Periodic Continued Fraction/Cycle/Length",
"Definition:Convergent of Continued Fraction"
] | [
"Definition:Positive/Integer",
"Solution of Pell's Equation is a Convergent",
"Definition:Continued Fraction",
"Definition:Periodic Continued Fraction",
"Continued Fraction Expansion of Irrational Square Root",
"Definition:Simple Continued Fraction",
"Definition:Continued Fraction/Finite",
"Definition... |
proofwiki-22147 | Induced Algebra Homomorphism on Unitization | Let $A, B$ be algebras over $\C$.
Let $\phi : A \to B$ be an algebra homomorphism.
Let $A_+$ and $B_+$ be the unitizations of $A$ and $B$ respectively.
Define $\psi : A_+ \to B_+$ by:
:$\map \psi {a, \lambda} = \tuple {\map \phi a, \lambda}$
for each $\tuple {a, \lambda} \in A_+$.
Then $\psi$ is a unital algebra homom... | Let $\tuple {a, \lambda}, \tuple {b, \mu} \in A_+$ and $t \in \C$.
We then have:
{{begin-eqn}}
{{eqn | l = \map \psi {\tuple {a, \lambda} + t \tuple {b, \mu} }
| r = \map \psi {\tuple {a + t b, \lambda + t \mu} }
}}
{{eqn | r = \tuple {\map \phi {a + t b}, \lambda + t \mu}
}}
{{eqn | r = \tuple {\map \phi a + t \map... | Let $A, B$ be [[Definition:Algebra over Field|algebras]] over $\C$.
Let $\phi : A \to B$ be an [[Definition:Algebra Homomorphism|algebra homomorphism]].
Let $A_+$ and $B_+$ be the [[Definition:Unitization of Algebra over Field|unitizations]] of $A$ and $B$ respectively.
Define $\psi : A_+ \to B_+$ by:
:$\map \psi {a... | Let $\tuple {a, \lambda}, \tuple {b, \mu} \in A_+$ and $t \in \C$.
We then have:
{{begin-eqn}}
{{eqn | l = \map \psi {\tuple {a, \lambda} + t \tuple {b, \mu} }
| r = \map \psi {\tuple {a + t b, \lambda + t \mu} }
}}
{{eqn | r = \tuple {\map \phi {a + t b}, \lambda + t \mu}
}}
{{eqn | r = \tuple {\map \phi a + t \ma... | Induced Algebra Homomorphism on Unitization | https://proofwiki.org/wiki/Induced_Algebra_Homomorphism_on_Unitization | https://proofwiki.org/wiki/Induced_Algebra_Homomorphism_on_Unitization | [
"Unitizations of Algebras over Fields"
] | [
"Definition:Algebra over Field",
"Definition:Algebra Homomorphism",
"Definition:Unitization of Algebra over Field",
"Definition:Unital Algebra Homomorphism"
] | [
"Definition:Unital Algebra Homomorphism",
"Category:Unitizations of Algebras over Fields"
] |
proofwiki-22148 | Spectrum of Element in Unitization of Unital Algebra | Let $A$ be a unital algebra over $\C$.
Let $A_+$ be the unitization of $A$.
Let $a \in A$.
Then:
:$\map {\sigma_{A_+} } {\tuple {a, 0} } = \map {\sigma_A} a \cup \set 0$
where $\sigma_{A_+}$ and $\sigma_A$ denote the spectrum in $A_+$ and $A$ respectively. | First, we note that $0 \in \map {\sigma_{A_+} } {\tuple {a, 0} }$.
Note that for each $\tuple {b, \lambda} \in A_+$, we have $\tuple {a, 0} \tuple {b, \lambda} = \tuple {a b + \lambda a, 0}$.
This cannot be equal to $\tuple { {\mathbf 0}_A, 1}$.
Hence $\tuple {a, 0}$ is not invertible and $0 \in \map {\sigma_{A_+} } {\... | Let $A$ be a [[Definition:Unital Algebra|unital]] [[Definition:Algebra over Field|algebra]] over $\C$.
Let $A_+$ be the [[Definition:Unitization of Algebra over Field|unitization]] of $A$.
Let $a \in A$.
Then:
:$\map {\sigma_{A_+} } {\tuple {a, 0} } = \map {\sigma_A} a \cup \set 0$
where $\sigma_{A_+}$ and $\sigma... | First, we note that $0 \in \map {\sigma_{A_+} } {\tuple {a, 0} }$.
Note that for each $\tuple {b, \lambda} \in A_+$, we have $\tuple {a, 0} \tuple {b, \lambda} = \tuple {a b + \lambda a, 0}$.
This cannot be equal to $\tuple { {\mathbf 0}_A, 1}$.
Hence $\tuple {a, 0}$ is not [[Definition:Invertible Element|invertible... | Spectrum of Element in Unitization of Unital Algebra | https://proofwiki.org/wiki/Spectrum_of_Element_in_Unitization_of_Unital_Algebra | https://proofwiki.org/wiki/Spectrum_of_Element_in_Unitization_of_Unital_Algebra | [
"Unitizations of Algebras over Fields",
"Spectra (Spectral Theory)"
] | [
"Definition:Unital Algebra",
"Definition:Algebra over Field",
"Definition:Unitization of Algebra over Field",
"Definition:Spectrum (Spectral Theory)"
] | [
"Definition:Invertible Element",
"Definition:Invertible Element",
"Definition:Invertible Element",
"Definition:Invertible Element",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Invertible Element",
"Definition:Invertible Element",
"Definition:Invertible Element",
"Definition:Inverse ... |
proofwiki-22149 | Bound on Inverse in Canonical Preordering of Unital C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $\le_A$ be the canonical preordering on $A$.
Let $a \in A$ and $\alpha, \beta > 0$ be such that:
:$\alpha {\mathbf 1}_A \le_A a \le_A \beta {\mathbf 1}_A$
Then:
:$\beta^{-1} {\mathbf 1}_A \le_A a^{-1} \le_A \alpha^{-1} {\mathbf 1}_A$ | Since $\beta^{-1} {\mathbf 1}_A \le_A a$ and ${\mathbf 0}_A \le_A \beta^{-1} {\mathbf 1}_A$, we have:
:$a$ is positive
since $\le_A$ is a preordering.
In particular, $a$ is Hermitian.
From Bounds on Hermitian Element of Unital C*-Algebra in terms of Bounds on Spectrum, we have:
:$\map {\sigma_A} a \subseteq \closedint ... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\le_A$ be the [[Definition:Canonical Preordering of C*-Algebra|canonical preordering]] on $A$.
Let $a \in A$ and $\alpha, \beta > 0$ be such that:
:$\alpha {\mathbf 1}_A \le_A a \le_A \beta {\mathbf 1}_A$
Then:... | Since $\beta^{-1} {\mathbf 1}_A \le_A a$ and ${\mathbf 0}_A \le_A \beta^{-1} {\mathbf 1}_A$, we have:
:$a$ is [[Definition:Positive Element of C*-Algebra|positive]]
since $\le_A$ is a [[Definition:Preordering|preordering]].
In particular, $a$ is [[Definition:Hermitian Element of *-Algebra|Hermitian]].
From [[Bounds o... | Bound on Inverse in Canonical Preordering of Unital C*-Algebra | https://proofwiki.org/wiki/Bound_on_Inverse_in_Canonical_Preordering_of_Unital_C*-Algebra | https://proofwiki.org/wiki/Bound_on_Inverse_in_Canonical_Preordering_of_Unital_C*-Algebra | [
"Canonical Preorderings of C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Canonical Preordering of C*-Algebra"
] | [
"Definition:Positive Element of C*-Algebra",
"Definition:Preordering",
"Definition:Hermitian Element of *-Algebra",
"Bounds on Hermitian Element of Unital C*-Algebra in terms of Bounds on Spectrum",
"Definition:Invertible Element",
"Spectrum of Inverse of Element of Unital Algebra",
"Bounds on Hermitian... |
proofwiki-22150 | Spectrum of Inverse of Element of Unital Algebra | Let $A$ be an algebra over $\C$.
Let $a \in A$ be invertible.
Let $\sigma_A$ denote the spectrum in $A$.
Then we have:
:$\map {\sigma_A} {a^{-1} } = \set {\lambda^{-1} : \lambda \in \map {\sigma_A} a}$ | Since $a$ is invertible, we have $0 \not \in \map {\sigma_A} a$ and $0 \not \in \map {\sigma_A} {a^{-1} }$.
We show that for $\lambda \in \C \setminus \set 0$:
:$\lambda {\mathbf 1}_A - a$ is invertible {{iff}} $\lambda^{-1} {\mathbf 1}_A - a^{-1}$ is invertible.
We can write:
:$\lambda^{-1} {\mathbf 1}_A - a^{-1} = a^... | Let $A$ be an [[Definition:Algebra over Field|algebra]] over $\C$.
Let $a \in A$ be [[Definition:Invertible Element|invertible]].
Let $\sigma_A$ denote the [[Definition:Spectrum (Spectral Theory)/Unital Algebra|spectrum]] in $A$.
Then we have:
:$\map {\sigma_A} {a^{-1} } = \set {\lambda^{-1} : \lambda \in \map {\si... | Since $a$ is [[Definition:Invertible Element|invertible]], we have $0 \not \in \map {\sigma_A} a$ and $0 \not \in \map {\sigma_A} {a^{-1} }$.
We show that for $\lambda \in \C \setminus \set 0$:
:$\lambda {\mathbf 1}_A - a$ is [[Definition:Invertible Element|invertible]] {{iff}} $\lambda^{-1} {\mathbf 1}_A - a^{-1}$ is... | Spectrum of Inverse of Element of Unital Algebra | https://proofwiki.org/wiki/Spectrum_of_Inverse_of_Element_of_Unital_Algebra | https://proofwiki.org/wiki/Spectrum_of_Inverse_of_Element_of_Unital_Algebra | [
"Spectra (Spectral Theory)"
] | [
"Definition:Algebra over Field",
"Definition:Invertible Element",
"Definition:Spectrum (Spectral Theory)/Unital Algebra"
] | [
"Definition:Invertible Element",
"Definition:Invertible Element",
"Definition:Invertible Element",
"Definition:Invertible Element",
"Definition:Invertible Element",
"Inverse of Product",
"Definition:Invertible Element",
"Inverse of Product",
"Definition:Invertible Element",
"Definition:Invertible ... |
proofwiki-22151 | Inverse reverses Preorder of Positive Invertible Elements of Unital C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $\le_A$ be the canonical preordering on $A$.
Let $a, b \in A$ be positive and invertible such that:
:$a \le_A b$
Then we have:
:$0 \le_A b^{-1} \le_A a^{-1}$ | First suppose that $c \in A$ satisfies:
:$c \ge_A {\mathbf 1}_A$
From Bounds on Hermitian Element of Unital C*-Algebra in terms of Bounds on Spectrum, we have:
:$c \le_A \norm c {\mathbf 1}_A$
From Bound on Inverse in Canonical Preordering of Unital C*-Algebra we have:
:$\norm c^{-1} {\mathbf 1}_A \le_A c^{-1} \le_A {... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\le_A$ be the [[Definition:Canonical Preordering of C*-Algebra|canonical preordering]] on $A$.
Let $a, b \in A$ be [[Definition:Positive Element of C*-Algebra|positive]... | First suppose that $c \in A$ satisfies:
:$c \ge_A {\mathbf 1}_A$
From [[Bounds on Hermitian Element of Unital C*-Algebra in terms of Bounds on Spectrum]], we have:
:$c \le_A \norm c {\mathbf 1}_A$
From [[Bound on Inverse in Canonical Preordering of Unital C*-Algebra]] we have:
:$\norm c^{-1} {\mathbf 1}_A \le_A c^{-... | Inverse reverses Preorder of Positive Invertible Elements of Unital C*-Algebra | https://proofwiki.org/wiki/Inverse_reverses_Preorder_of_Positive_Invertible_Elements_of_Unital_C*-Algebra | https://proofwiki.org/wiki/Inverse_reverses_Preorder_of_Positive_Invertible_Elements_of_Unital_C*-Algebra | [
"Positive Elements of C*-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Canonical Preordering of C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Invertible Element"
] | [
"Bounds on Hermitian Element of Unital C*-Algebra in terms of Bounds on Spectrum",
"Bound on Inverse in Canonical Preordering of Unital C*-Algebra",
"Spectrum of Inverse of Element of Unital Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Nth Root of Positive Element of C*-Algebra",
"De... |
proofwiki-22152 | Element of Unital C*-Algebra Bounded by Multiple of Identity is Hermitian | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $a \in A$.
Let $\le_A$ be the canonical preordering of $A$.
Suppose that either:
:$a \le_A \beta {\mathbf 1}_A$ for some $\alpha \in \R$
or:
:$\alpha {\mathbf 1}_A \le_A a$ for some $\alpha \in \R$
Then $a$ is Hermitian. | Suppose first that:
:$a \le_A \beta {\mathbf 1}_A$ for some $\alpha \in \R$
Then $\beta {\mathbf 1}_A - a$ is positive.
In particular, $\beta {\mathbf 1}_A - a$ is Hermitian.
That is:
:$\paren {\beta {\mathbf 1}_A - a}^\ast = \overline \beta {\mathbf 1}_A - a^\ast = \beta {\mathbf 1}_A - a$
from the definition of an in... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $a \in A$.
Let $\le_A$ be the [[Definition:Canonical Preordering of C*-Algebra|canonical preordering]] of $A$.
Suppose that either:
:$a \le_A \beta {\mathbf 1}_A$ for s... | Suppose first that:
:$a \le_A \beta {\mathbf 1}_A$ for some $\alpha \in \R$
Then $\beta {\mathbf 1}_A - a$ is [[Definition:Positive Element of C*-Algebra|positive]].
In particular, $\beta {\mathbf 1}_A - a$ is [[Definition:Hermitian Element of *-Algebra|Hermitian]].
That is:
:$\paren {\beta {\mathbf 1}_A - a}^\ast =... | Element of Unital C*-Algebra Bounded by Multiple of Identity is Hermitian | https://proofwiki.org/wiki/Element_of_Unital_C*-Algebra_Bounded_by_Multiple_of_Identity_is_Hermitian | https://proofwiki.org/wiki/Element_of_Unital_C*-Algebra_Bounded_by_Multiple_of_Identity_is_Hermitian | [
"Canonical Preorderings of C*-Algebras",
"Hermitian Elements of *-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Canonical Preordering of C*-Algebra",
"Definition:Hermitian Element of *-Algebra"
] | [
"Definition:Positive Element of C*-Algebra",
"Definition:Hermitian Element of *-Algebra",
"Definition:Involution on Algebra",
"Identity Element in Unital *-Algebra is Hermitian",
"Definition:Hermitian Element of *-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Hermitian Element of *-... |
proofwiki-22153 | Element of Unital C*-Algebra Bounded below by Non-Negative Multiple of Identity Element is Invertible and Positive | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $\le_A$ be the canonical preordering of $A$.
Let $a \in A$ be such that:
:$a \ge_A \alpha {\mathbf 1}_A$
for some $\alpha > 0$.
Then $a$ is invertible and positive element. | From Element of Unital C*-Algebra Bounded by Multiple of Identity is Hermitian, $a$ is Hermitian.
From Bounds on Hermitian Element of Unital C*-Algebra in terms of Bounds on Spectrum, we then have:
:$\map {\sigma_A} a \subseteq \hointr \alpha \infty \subseteq \hointr 0 \infty$
So $a$ is positive.
We have $\alpha > 0$ ... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\le_A$ be the [[Definition:Canonical Preordering of C*-Algebra|canonical preordering]] of $A$.
Let $a \in A$ be such that:
:$a \ge_A \alpha {\mathbf 1}_A$
for some $\alpha > 0... | From [[Element of Unital C*-Algebra Bounded by Multiple of Identity is Hermitian]], $a$ is [[Definition:Hermitian Element of *-Algebra|Hermitian]].
From [[Bounds on Hermitian Element of Unital C*-Algebra in terms of Bounds on Spectrum]], we then have:
:$\map {\sigma_A} a \subseteq \hointr \alpha \infty \subseteq \hoin... | Element of Unital C*-Algebra Bounded below by Non-Negative Multiple of Identity Element is Invertible and Positive | https://proofwiki.org/wiki/Element_of_Unital_C*-Algebra_Bounded_below_by_Non-Negative_Multiple_of_Identity_Element_is_Invertible_and_Positive | https://proofwiki.org/wiki/Element_of_Unital_C*-Algebra_Bounded_below_by_Non-Negative_Multiple_of_Identity_Element_is_Invertible_and_Positive | [
"Canonical Preorderings of C*-Algebras"
] | [
"Definition:Unital Algebra",
"Definition:C*-Algebra",
"Definition:Canonical Preordering of C*-Algebra",
"Definition:Invertible Element",
"Definition:Positive Element of C*-Algebra"
] | [
"Element of Unital C*-Algebra Bounded by Multiple of Identity is Hermitian",
"Definition:Hermitian Element of *-Algebra",
"Bounds on Hermitian Element of Unital C*-Algebra in terms of Bounds on Spectrum",
"Definition:Positive Element of C*-Algebra",
"Definition:Invertible Element",
"Category:Canonical Pre... |
proofwiki-22154 | Spectrum of Identity Element of Unital Algebra | Let $A$ be a unital algebra over $\C$ with at least two distinct elements.
Let $\sigma_A$ be the spectrum in $A$.
Then we have:
:$\map {\sigma_A} { {\mathbf 1}_A} = \set 1$ | We have:
:${\mathbf 1}_A - {\mathbf 1}_A = {\mathbf 0}_A$
which is not invertible.
So $1 \in \map {\sigma_A} { {\mathbf 1}_A}$.
Now suppose that $\lambda \in \C \setminus \set 1$.
Then we have:
:$\lambda {\mathbf 1}_A - {\mathbf 1}_A = \paren {\lambda - 1} {\mathbf 1}_A$.
Hence:
:$\paren {\paren {\lambda - 1} {\mathbf... | Let $A$ be a [[Definition:Unital Algebra|unital algebra]] over $\C$ with at least two distinct elements.
Let $\sigma_A$ be the [[Definition:Spectrum (Spectral Theory)|spectrum]] in $A$.
Then we have:
:$\map {\sigma_A} { {\mathbf 1}_A} = \set 1$ | We have:
:${\mathbf 1}_A - {\mathbf 1}_A = {\mathbf 0}_A$
which is not [[Definition:Invertible Element|invertible]].
So $1 \in \map {\sigma_A} { {\mathbf 1}_A}$.
Now suppose that $\lambda \in \C \setminus \set 1$.
Then we have:
:$\lambda {\mathbf 1}_A - {\mathbf 1}_A = \paren {\lambda - 1} {\mathbf 1}_A$.
Hence:
:... | Spectrum of Identity Element of Unital Algebra | https://proofwiki.org/wiki/Spectrum_of_Identity_Element_of_Unital_Algebra | https://proofwiki.org/wiki/Spectrum_of_Identity_Element_of_Unital_Algebra | [
"Spectra (Spectral Theory)"
] | [
"Definition:Unital Algebra",
"Definition:Spectrum (Spectral Theory)"
] | [
"Definition:Invertible Element",
"Definition:Invertible Element",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Category:Spectra (Spectral Theory)"
] |
proofwiki-22155 | Multiple of Identity Element of Unital C*-Algebra is Positive iff Coefficient is Non-Negative Real Number | Let $\struct {A, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $\lambda \in \C$.
Then $\lambda {\mathbf 1}_A$ is positive {{iff}} $\lambda \in \R_{\ge 0}$. | First we have:
:$\tuple {\lambda {\mathbf 1}_A}^\ast = \overline \lambda {\mathbf 1}_A$
by the definition of an involution and Identity Element in Unital *-Algebra is Hermitian.
Since:
:$\lambda \overline \lambda {\mathbf 1}_A = \overline \lambda \lambda {\mathbf 1}_A$
Hence $\lambda {\mathbf 1}_A$ is normal.
From the ... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\lambda \in \C$.
Then $\lambda {\mathbf 1}_A$ is [[Definition:Positive Element of C*-Algebra|positive]] {{iff}} $\lambda \in \R_{\ge 0}$. | First we have:
:$\tuple {\lambda {\mathbf 1}_A}^\ast = \overline \lambda {\mathbf 1}_A$
by the definition of an [[Definition:Involution on Algebra|involution]] and [[Identity Element in Unital *-Algebra is Hermitian]].
Since:
:$\lambda \overline \lambda {\mathbf 1}_A = \overline \lambda \lambda {\mathbf 1}_A$
Hence $... | Multiple of Identity Element of Unital C*-Algebra is Positive iff Coefficient is Non-Negative Real Number | https://proofwiki.org/wiki/Multiple_of_Identity_Element_of_Unital_C*-Algebra_is_Positive_iff_Coefficient_is_Non-Negative_Real_Number | https://proofwiki.org/wiki/Multiple_of_Identity_Element_of_Unital_C*-Algebra_is_Positive_iff_Coefficient_is_Non-Negative_Real_Number | [
"Positive Elements of C*-Algebras",
"Multiple of Identity Element of Unital C*-Algebra is Positive iff Coefficient is Non-Negative Real Number"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Positive Element of C*-Algebra"
] | [
"Definition:Involution on Algebra",
"Identity Element in Unital *-Algebra is Hermitian",
"Definition:Normal Element of *-Algebra",
"Spectral Mapping Theorem for Polynomials",
"Spectrum of Identity Element of Unital Algebra",
"Normal Element of C*-Algebra is Hermitian iff Spectrum is Real",
"Definition:H... |
proofwiki-22156 | Multiple of Identity Element of Unital C*-Algebra is Positive iff Coefficient is Non-Negative Real Number/Corollary | Let $\struct {A, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $\lambda, \mu \in \R$.
Let $\le_A$ be the canonical preordering of $A$.
Then $\lambda {\mathbf 1}_A \le_A \mu {\mathbf 1}_A$ {{iff}} $\lambda \le \mu$. | We have $\lambda {\mathbf 1}_A \le_A \mu {\mathbf 1}_A$ {{iff}} $\paren {\mu - \lambda} {\mathbf 1}_A$ is positive.
From Multiple of Identity Element of Unital C*-Algebra is Positive iff Coefficient is Non-Negative Real Number, we have that $\paren {\mu - \lambda} {\mathbf 1}_A$ is positive {{iff}} $\mu - \lambda \ge 0... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\lambda, \mu \in \R$.
Let $\le_A$ be the [[Definition:Canonical Preordering of C*-Algebra|canonical preordering]] of $A$.
Then $\lambda {\mathbf 1}_A \le_A \mu {\mathbf 1}_... | We have $\lambda {\mathbf 1}_A \le_A \mu {\mathbf 1}_A$ {{iff}} $\paren {\mu - \lambda} {\mathbf 1}_A$ is [[Definition:Positive Element of C*-Algebra|positive]].
From [[Multiple of Identity Element of Unital C*-Algebra is Positive iff Coefficient is Non-Negative Real Number]], we have that $\paren {\mu - \lambda} {\ma... | Multiple of Identity Element of Unital C*-Algebra is Positive iff Coefficient is Non-Negative Real Number/Corollary | https://proofwiki.org/wiki/Multiple_of_Identity_Element_of_Unital_C*-Algebra_is_Positive_iff_Coefficient_is_Non-Negative_Real_Number/Corollary | https://proofwiki.org/wiki/Multiple_of_Identity_Element_of_Unital_C*-Algebra_is_Positive_iff_Coefficient_is_Non-Negative_Real_Number/Corollary | [
"Multiple of Identity Element of Unital C*-Algebra is Positive iff Coefficient is Non-Negative Real Number"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Canonical Preordering of C*-Algebra"
] | [
"Definition:Positive Element of C*-Algebra",
"Multiple of Identity Element of Unital C*-Algebra is Positive iff Coefficient is Non-Negative Real Number",
"Definition:Positive Element of C*-Algebra",
"Category:Multiple of Identity Element of Unital C*-Algebra is Positive iff Coefficient is Non-Negative Real Nu... |
proofwiki-22157 | Square Root is Increasing with respect to Canonical Preordering of C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $\le_A$ be the canonical preordering of $A$.
Let $x, y \in A$ be positive and such that:
:$x \le_A y$
Let $x^{1/2}$ and $y^{1/2}$ be the positive square roots of $a$ and $b$ respectively.
Then:
:$x^{1/2} \le_A y^{1/2}$ | Write $a = x^{1/2}$ and $y^{1/2}$.
We then have $a^2 \le_A b^2$ and want to show that $a \le_A b$.
First take $A$ unital.
Let $t > 0$.
Let:
:$u = \paren {t {\mathbf 1}_A + b + a} \paren {t {\mathbf 1}_A + b - a}$
From Element of *-Algebra Uniquely Decomposes into Hermitian Elements, there exists unique Hermitian ele... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\le_A$ be the [[Definition:Canonical Preordering of C*-Algebra|canonical preordering]] of $A$.
Let $x, y \in A$ be [[Definition:Positive Element of C*-Algebra|positive]] and such that:
:$x \le_A y$
Let $x^{1/2}... | Write $a = x^{1/2}$ and $y^{1/2}$.
We then have $a^2 \le_A b^2$ and want to show that $a \le_A b$.
First take $A$ [[Definition:Unital Banach Algebra|unital]].
Let $t > 0$.
Let:
:$u = \paren {t {\mathbf 1}_A + b + a} \paren {t {\mathbf 1}_A + b - a}$
From [[Element of *-Algebra Uniquely Decomposes into Hermitian... | Square Root is Increasing with respect to Canonical Preordering of C*-Algebra | https://proofwiki.org/wiki/Square_Root_is_Increasing_with_respect_to_Canonical_Preordering_of_C*-Algebra | https://proofwiki.org/wiki/Square_Root_is_Increasing_with_respect_to_Canonical_Preordering_of_C*-Algebra | [
"Positive Elements of C*-Algebras",
"Canonical Preorderings of C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Canonical Preordering of C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Nth Root of Positive Element of C*-Algebra"
] | [
"Definition:Unital Banach Algebra",
"Element of *-Algebra Uniquely Decomposes into Hermitian Elements",
"Definition:Hermitian Element of *-Algebra",
"Definition:Involution on Algebra",
"Identity Element in Unital *-Algebra is Hermitian",
"Non-Negative Multiple of Positive Element of C*-Algebra is Positive... |
proofwiki-22158 | Hermitian Element of C*-Algebra Decomposes into Positive Elements | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $b \in A$ be Hermitian.
Then there exists positive elements $b^+$ and $b^-$ such that:
:$b^+ b^- = {\mathbf 0}_A$
:$b^+ - b^- = b$
:$\norm {b^+} \le \norm b$ and $\norm {b^-} \le \norm b$. | First let $A$ be unital.
Let $B$ be be the $\text C^\ast$-algebra generated by $\set { {\mathbf 1}_A, b}$.
Let $\Phi_B$ be the spectrum of $B$.
By C*-Algebra Generated by Commutative Self-Adjoint Set is Commutative, $B$ is commutative.
Let $\struct {\map {\CC_0} {\Phi_B}, \overline \cdot, \norm {\, \cdot \,}_\infty}$ ... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $b \in A$ be [[Definition:Hermitian Element of *-Algebra|Hermitian]].
Then there exists [[Definition:Positive Element of C*-Algebra|positive elements]] $b^+$ and $b^-$ such that:
:$b^+ b^- = {\mathbf 0}_A$
:$b^+... | First let $A$ be [[Definition:Unital Algebra|unital]].
Let $B$ be be the [[Definition:Generated C*-Algebra|$\text C^\ast$-algebra generated]] by $\set { {\mathbf 1}_A, b}$.
Let $\Phi_B$ be the [[Definition:Spectrum of Banach Algebra|spectrum]] of $B$.
By [[C*-Algebra Generated by Commutative Self-Adjoint Set is Com... | Hermitian Element of C*-Algebra Decomposes into Positive Elements | https://proofwiki.org/wiki/Hermitian_Element_of_C*-Algebra_Decomposes_into_Positive_Elements | https://proofwiki.org/wiki/Hermitian_Element_of_C*-Algebra_Decomposes_into_Positive_Elements | [
"Positive Elements of C*-Algebras",
"Hermitian Element of C*-Algebra Decomposes into Positive Elements"
] | [
"Definition:C*-Algebra",
"Definition:Hermitian Element of *-Algebra",
"Definition:Positive Element of C*-Algebra"
] | [
"Definition:Unital Algebra",
"Definition:Generated C*-Algebra",
"Definition:Spectrum of Banach Algebra",
"C*-Algebra Generated by Commutative Self-Adjoint Set is Commutative",
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:C*-Algebra",
"Definition:Continuous Mapping",
"Definition:Com... |
proofwiki-22159 | Element of C*-Algebra is Positive iff Positive in Unitization | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $A_+$ be the unitization of $A$.
Define:
:$\tuple {x, \lambda}^\ast = \tuple {x^\ast, \overline \lambda}$
for each $\tuple {x, \lambda} \in A_+$.
Let $\norm {\, \cdot \,}_\ast$ be such that $\struct {A_+, \ast, \norm {\, \cdot \,}_\ast}$ is... | The existence of $\norm {\, \cdot \,}_\ast$ follows from:
:Existence of Unitization of C*-Algebra if $A$ is non-unital
:Existence of Unique C* Norm on Unitization of Unital C*-Algebra if $A$ is unital
From Element of *-Algebra is Hermitian iff Hermitian in Unitization:
:$a$ is Hermitian in $A$ {{iff}} $\tuple {a, 0}$ i... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $A_+$ be the [[Definition:Unitization of Algebra over Field|unitization]] of $A$.
Define:
:$\tuple {x, \lambda}^\ast = \tuple {x^\ast, \overline \lambda}$
for each $\tuple {x, \lambda} \in A_+$.
Let $\norm {\, \... | The existence of $\norm {\, \cdot \,}_\ast$ follows from:
:[[Existence of Unitization of C*-Algebra]] if $A$ is non-[[Definition:Unital Banach Algebra|unital]]
:[[Existence of Unique C* Norm on Unitization of Unital C*-Algebra]] if $A$ is [[Definition:Unital Banach Algebra|unital]]
From [[Element of *-Algebra is Hermi... | Element of C*-Algebra is Positive iff Positive in Unitization | https://proofwiki.org/wiki/Element_of_C*-Algebra_is_Positive_iff_Positive_in_Unitization | https://proofwiki.org/wiki/Element_of_C*-Algebra_is_Positive_iff_Positive_in_Unitization | [
"Element of C*-Algebra is Positive iff Positive in Unitization",
"Positive Elements of C*-Algebras",
"Element of C*-Algebra is Positive iff Positive in Unitization"
] | [
"Definition:C*-Algebra",
"Definition:Unitization of Algebra over Field",
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Element of C*-Algebra"
] | [
"Existence of Unitization of C*-Algebra",
"Definition:Unital Banach Algebra",
"Existence of Unique C* Norm on Unitization of Unital C*-Algebra",
"Definition:Unital Banach Algebra",
"Element of *-Algebra is Hermitian iff Hermitian in Unitization",
"Definition:Hermitian Element of *-Algebra",
"Definition:... |
proofwiki-22160 | Continuous Function applied to Normal Element of Unital C*-Algebra is Positive iff Function is Non-Negative/Corollary | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $x \in A$ be normal.
Let $\map {\sigma_A} x$ denote the spectrum of $x$ in $A$.
Let $f, g, h : \map {\sigma_A} x \to \R$ be continuous such that:
:$\map g z \le \map f z$ for all $z \in \map {\sigma_A} x$.
Let $\map {\Theta_x} f = \... | Since $g \le f$ we have:
:$\map {\paren {f - g} } z \in \R_{\ge 0}$ for all $z \in \map {\sigma_A} x$.
Hence from Continuous Function applied to Normal Element of Unital C*-Algebra is Positive iff Function is Non-Negative, we have:
:$\map {\Theta_x} {f - g} \ge_A {\mathbf 0}_A$
Since $\Theta_x$ is linear, we have:
:$\m... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $x \in A$ be [[Definition:Normal Element of *-Algebra|normal]].
Let $\map {\sigma_A} x$ denote the [[Definition:Spectrum (Spectral Theory)/Unital Algebra|spectrum]] of ... | Since $g \le f$ we have:
:$\map {\paren {f - g} } z \in \R_{\ge 0}$ for all $z \in \map {\sigma_A} x$.
Hence from [[Continuous Function applied to Normal Element of Unital C*-Algebra is Positive iff Function is Non-Negative]], we have:
:$\map {\Theta_x} {f - g} \ge_A {\mathbf 0}_A$
Since $\Theta_x$ is [[Definition:Li... | Continuous Function applied to Normal Element of Unital C*-Algebra is Positive iff Function is Non-Negative/Corollary | https://proofwiki.org/wiki/Continuous_Function_applied_to_Normal_Element_of_Unital_C*-Algebra_is_Positive_iff_Function_is_Non-Negative/Corollary | https://proofwiki.org/wiki/Continuous_Function_applied_to_Normal_Element_of_Unital_C*-Algebra_is_Positive_iff_Function_is_Non-Negative/Corollary | [
"Continuous Function applied to Normal Element of Unital C*-Algebra is Positive iff Function is Non-Negative"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Normal Element of *-Algebra",
"Definition:Spectrum (Spectral Theory)/Unital Algebra",
"Definition:Continuous Function",
"Definition:Continuous Functional Calculus",
"Definition:Canonical Preordering of C*-Algebra"
] | [
"Continuous Function applied to Normal Element of Unital C*-Algebra is Positive iff Function is Non-Negative",
"Definition:Linear Transformation",
"Category:Continuous Function applied to Normal Element of Unital C*-Algebra is Positive iff Function is Non-Negative"
] |
proofwiki-22161 | Bound for Norm of Difference of Element with Non-Negative Multiple of Element in Unital C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $\le_A$ be the canonical preordering of $A$.
Let $a, b \in A$ be such that:
:${\mathbf 0}_A \le_A a \le_A b \le_A {\mathbf 1}_A$
Then we have:
:$\norm {x - b x}^2 \le \norm {x^\ast \paren { {\mathbf 1}_A - a} x}$
and:
:$\norm {x - x ... | We have from Scalar Multiplication by Minus One reverses Preordering in Vector Space:
:$-{\mathbf 1}_A \le_A -b \le_A -a \le_A {\mathbf 0}_A$
From $(1)$ in the definition of a preordered vector space, we have:
:${\mathbf 1}_A - {\mathbf 1}_A \le_A {\mathbf 1}_A - b \le_A {\mathbf 1}_A$
so that:
:${\mathbf 0}_A \le_A {\... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\le_A$ be the [[Definition:Canonical Preordering of C*-Algebra|canonical preordering]] of $A$.
Let $a, b \in A$ be such that:
:${\mathbf 0}_A \le_A a \le_A b \le_A {\m... | We have from [[Scalar Multiplication by Minus One reverses Preordering in Vector Space]]:
:$-{\mathbf 1}_A \le_A -b \le_A -a \le_A {\mathbf 0}_A$
From $(1)$ in the definition of a [[Definition:Preordered Vector Space|preordered vector space]], we have:
:${\mathbf 1}_A - {\mathbf 1}_A \le_A {\mathbf 1}_A - b \le_A {\ma... | Bound for Norm of Difference of Element with Non-Negative Multiple of Element in Unital C*-Algebra | https://proofwiki.org/wiki/Bound_for_Norm_of_Difference_of_Element_with_Non-Negative_Multiple_of_Element_in_Unital_C*-Algebra | https://proofwiki.org/wiki/Bound_for_Norm_of_Difference_of_Element_with_Non-Negative_Multiple_of_Element_in_Unital_C*-Algebra | [
"C*-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Canonical Preordering of C*-Algebra"
] | [
"Scalar Multiplication by Minus One reverses Preordering in Vector Space",
"Definition:Preordered Vector Space",
"Bounds on Hermitian Element of Unital C*-Algebra in terms of Bounds on Spectrum",
"Continuous Function applied to Normal Element of Unital C*-Algebra is Positive iff Function is Non-Negative/Corol... |
proofwiki-22162 | Vector Space obtains Norm Structure through Linear Isomorphism | Let $\GF \in \set {\R, \C}$.
Let $X$ and $Y$ be vector spaces over $\GF$.
Let $\norm {\, \cdot \,}_Y$ be a norm on $Y$.
Let $\phi : X \to Y$ be a linear isomorphism.
Define $\norm {\, \cdot \,}_X : X \to \R$ by:
:$\norm x_X = \norm {\map \phi x}_Y$
for each $x \in X$.
Then $\norm {\, \cdot \,}_X$ is a norm on $X$. | First since $\norm y_Y \in \hointr 0 \infty$ for each $y \in Y$, we have $\norm {\map \phi x}_Y \in \hointr 0 \infty$ for each $x \in X$. | Let $\GF \in \set {\R, \C}$.
Let $X$ and $Y$ be [[Definition:Vector Space|vector spaces]] over $\GF$.
Let $\norm {\, \cdot \,}_Y$ be a [[Definition:Norm on Vector Space|norm]] on $Y$.
Let $\phi : X \to Y$ be a [[Definition:Linear Isomorphism|linear isomorphism]].
Define $\norm {\, \cdot \,}_X : X \to \R$ by:
:$\no... | First since $\norm y_Y \in \hointr 0 \infty$ for each $y \in Y$, we have $\norm {\map \phi x}_Y \in \hointr 0 \infty$ for each $x \in X$. | Vector Space obtains Norm Structure through Linear Isomorphism | https://proofwiki.org/wiki/Vector_Space_obtains_Norm_Structure_through_Linear_Isomorphism | https://proofwiki.org/wiki/Vector_Space_obtains_Norm_Structure_through_Linear_Isomorphism | [
"Normed Vector Spaces"
] | [
"Definition:Vector Space",
"Definition:Norm/Vector Space",
"Definition:Linear Isomorphism",
"Definition:Norm/Vector Space"
] | [] |
proofwiki-22163 | Algebra obtains Norm Structure through Algebra Isomorphism | Let $\GF \in \set {\R, \C}$.
Let $A$ and $B$ be algebras over $\GF$.
Let $\norm {\, \cdot \,}_B$ be an algebra norm on $B$.
Let $\phi : A \to B$ be an algebra isomorphism.
Define $\norm {\, \cdot \,}_A : A \to \R$ by:
:$\norm a_A = \norm {\map \phi a}_B$
Then $\norm {\, \cdot \,}_A$ is an algebra norm on $A$. | From Vector Space obtains Norm Structure through Linear Isomorphism:
:$\norm {\, \cdot \,}$ is a vector space norm on $A$.
We want to show that:
:$\norm {a b}_A \le \norm a_A \norm b_A$
for each $a, b \in A$.
Let $a, b \in A$.
We have:
{{begin-eqn}}
{{eqn | l = \norm {a b}_A
| r = \norm {\map \phi {a b} }_B
}}
{{eqn... | Let $\GF \in \set {\R, \C}$.
Let $A$ and $B$ be [[Definition:Algebra over Field|algebras]] over $\GF$.
Let $\norm {\, \cdot \,}_B$ be an [[Definition:Norm on Algebra|algebra norm]] on $B$.
Let $\phi : A \to B$ be an [[Definition:Algebra Isomorphism|algebra isomorphism]].
Define $\norm {\, \cdot \,}_A : A \to \R$ b... | From [[Vector Space obtains Norm Structure through Linear Isomorphism]]:
:$\norm {\, \cdot \,}$ is a [[Definition:Norm on Vector Space|vector space norm]] on $A$.
We want to show that:
:$\norm {a b}_A \le \norm a_A \norm b_A$
for each $a, b \in A$.
Let $a, b \in A$.
We have:
{{begin-eqn}}
{{eqn | l = \norm {a b}_A
... | Algebra obtains Norm Structure through Algebra Isomorphism | https://proofwiki.org/wiki/Algebra_obtains_Norm_Structure_through_Algebra_Isomorphism | https://proofwiki.org/wiki/Algebra_obtains_Norm_Structure_through_Algebra_Isomorphism | [
"Normed Algebras"
] | [
"Definition:Algebra over Field",
"Definition:Norm/Algebra",
"Definition:Algebra Isomorphism",
"Definition:Norm/Algebra"
] | [
"Vector Space obtains Norm Structure through Linear Isomorphism",
"Definition:Norm/Vector Space",
"Definition:Norm/Algebra",
"Category:Normed Algebras"
] |
proofwiki-22164 | *-Algebra obtains Banach *-Algebra Structure through *-Algebra Isomorphism | Let $\struct {A, \ast}$ be a $\ast$-algebra over $\C$.
Let $\struct {B, \square, \norm {\, \cdot \,}_B}$ be a Banach $\ast$-algebra.
Let $\phi : A \to B$ be a $\ast$-algebra isomorphism.
Define $\norm {\, \cdot \,}_A : A \to \R$ by:
:$\norm a_A = \norm {\map \phi a}_B$
Then $\struct {A, \ast, \norm {\, \cdot \,}_A}$ i... | From Algebra obtains Norm Structure through Algebra Isomorphism:
:$\norm {\, \cdot \,}_A$ is an algebra norm.
We want to show that:
:$\norm {a^\ast}_A = \norm a_A$
for each $a \in A$.
Let $a \in A$.
We have:
{{begin-eqn}}
{{eqn | l = \norm {a^\ast}_A
| r = \norm {\map \phi {a^\ast} }_B
}}
{{eqn | r = \norm {\map \p... | Let $\struct {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]] over $\C$.
Let $\struct {B, \square, \norm {\, \cdot \,}_B}$ be a [[Definition:Banach *-Algebra|Banach $\ast$-algebra]].
Let $\phi : A \to B$ be a [[Definition:*-Algebra Isomorphism|$\ast$-algebra isomorphism]].
Define $\norm {\, \cdot \,}_A : A \... | From [[Algebra obtains Norm Structure through Algebra Isomorphism]]:
:$\norm {\, \cdot \,}_A$ is an [[Definition:Norm on Algebra|algebra norm]].
We want to show that:
:$\norm {a^\ast}_A = \norm a_A$
for each $a \in A$.
Let $a \in A$.
We have:
{{begin-eqn}}
{{eqn | l = \norm {a^\ast}_A
| r = \norm {\map \phi {a^\a... | *-Algebra obtains Banach *-Algebra Structure through *-Algebra Isomorphism | https://proofwiki.org/wiki/*-Algebra_obtains_Banach_*-Algebra_Structure_through_*-Algebra_Isomorphism | https://proofwiki.org/wiki/*-Algebra_obtains_Banach_*-Algebra_Structure_through_*-Algebra_Isomorphism | [
"Banach *-Algebras",
"C*-Algebras"
] | [
"Definition:*-Algebra",
"Definition:Banach *-Algebra",
"Definition:*-Algebra Isomorphism",
"Definition:Banach *-Algebra",
"Definition:C*-Algebra"
] | [
"Algebra obtains Norm Structure through Algebra Isomorphism",
"Definition:Norm/Algebra",
"Definition:Isometric Isomorphism",
"Inverse of Isometric Isomorphism between Normed Vector Spaces is Isometric Isomorphism",
"Definition:Isometric Isomorphism",
"Metric Space Completeness is Preserved by Isometry",
... |
proofwiki-22165 | Direct Product of Algebras is Algebra | Let $K$ be a field.
Let $A$ and $B$ be algebras over $K$.
Let $\tuple {A \times B, +_{A \times B}, \cdot_{A \times B} }$ be the direct product of $A$ and $B$ as vector spaces.
Define $\cdot_{A \times B} : A \times B \to A \times B$ by:
:$\tuple {a, b} \tuple {a', b'} = \tuple {a a', b b'}$
for each $\tuple {a, b}, \tu... | We show that $\circ_{A \times B}$ is bilinear.
Let $\lambda \in K$ and $\tuple {a_1, b_1}, \tuple {a_2, b_2}, \tuple {a_3, b_3} \in A \times B$ .
We have:
{{begin-eqn}}
{{eqn | l = \paren {\tuple {a_1, b_1} +_{A \times B} \lambda \tuple {a_2, b_2} } \circ_{A \times B} \tuple {a_3, b_3}
| r = \tuple {a_1 + \lambda a_2... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $A$ and $B$ be [[Definition:Algebra over Field|algebras]] over $K$.
Let $\tuple {A \times B, +_{A \times B}, \cdot_{A \times B} }$ be the [[Definition:Direct Product of Vector Spaces|direct product]] of $A$ and $B$ as [[Definition:Vector Space|vector sp... | We show that $\circ_{A \times B}$ is [[Definition:Bilinear Mapping|bilinear]].
Let $\lambda \in K$ and $\tuple {a_1, b_1}, \tuple {a_2, b_2}, \tuple {a_3, b_3} \in A \times B$ .
We have:
{{begin-eqn}}
{{eqn | l = \paren {\tuple {a_1, b_1} +_{A \times B} \lambda \tuple {a_2, b_2} } \circ_{A \times B} \tuple {a_3, b_3}... | Direct Product of Algebras is Algebra | https://proofwiki.org/wiki/Direct_Product_of_Algebras_is_Algebra | https://proofwiki.org/wiki/Direct_Product_of_Algebras_is_Algebra | [
"Algebras over Fields"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Algebra over Field",
"Definition:Direct Product of Vector Spaces",
"Definition:Vector Space",
"Definition:Algebra over Field"
] | [
"Definition:Bilinear Mapping",
"Definition:Bilinear Mapping",
"Definition:Algebra over Field",
"Category:Algebras over Fields"
] |
proofwiki-22166 | Direct Product of Normed Algebras is Normed Algebra with Direct Product Norm | Let $\GF \in \set {\R, \C}$.
Let $\struct {A, \norm {\, \cdot \,}_A}$ and $\struct {B, \norm {\, \cdot \,}_B}$ be normed algebras over $\GF$.
Let $A \times B$ be the direct product of $A$ and $B$.
Let $\norm {\, \cdot \,}_{A \times B}$ be the direct product norm on $A \times B$.
Then $\struct {A \times B, \norm {\, \c... | From Direct Product Norm is Norm, $\struct {A \times B, \norm {\, \cdot \,}_{A \times B} }$ is a normed vector space.
We just need to show that $\norm {\, \cdot \,}_{A \times B}$ is an algebra norm.
Let $\tuple {a, b}, \tuple {a', b'} \in A \times B$.
We hae:
:$\norm {\tuple {a, b} \tuple {a', b'} }_{A \times B} = \no... | Let $\GF \in \set {\R, \C}$.
Let $\struct {A, \norm {\, \cdot \,}_A}$ and $\struct {B, \norm {\, \cdot \,}_B}$ be [[Definition:Normed Algebra|normed algebras]] over $\GF$.
Let $A \times B$ be the [[Definition:Direct Product of Algebras|direct product]] of $A$ and $B$.
Let $\norm {\, \cdot \,}_{A \times B}$ be the [... | From [[Direct Product Norm is Norm]], $\struct {A \times B, \norm {\, \cdot \,}_{A \times B} }$ is a [[Definition:Normed Vector Space|normed vector space]].
We just need to show that $\norm {\, \cdot \,}_{A \times B}$ is an [[Definition:Norm on Algebra|algebra norm]].
Let $\tuple {a, b}, \tuple {a', b'} \in A \times ... | Direct Product of Normed Algebras is Normed Algebra with Direct Product Norm | https://proofwiki.org/wiki/Direct_Product_of_Normed_Algebras_is_Normed_Algebra_with_Direct_Product_Norm | https://proofwiki.org/wiki/Direct_Product_of_Normed_Algebras_is_Normed_Algebra_with_Direct_Product_Norm | [
"Normed Algebras"
] | [
"Definition:Normed Algebra",
"Definition:Direct Product of Algebras",
"Definition:Direct Product Norm",
"Definition:Normed Algebra"
] | [
"Direct Product Norm is Norm",
"Definition:Normed Vector Space",
"Definition:Norm/Algebra",
"Definition:Norm/Algebra",
"Category:Normed Algebras"
] |
proofwiki-22167 | Direct Product of *-Algebras is *-Algebra | Let $\tuple {A, \square}$ and $\tuple {B, \diamond}$ be $\ast$-algebras over $\C$.
Let $A \times B$ be the direct product of $A$ and $B$.
Define $\ast : A \times B \to A \times B$ by:
:$\tuple {a, b}^\ast = \tuple {a^\square, b^\diamond}$
for each $\tuple {a, b} \in A \times B$.
Then $\tuple {A \times B, \ast}$ is a ... | We show that $\ast$ is a involution. | Let $\tuple {A, \square}$ and $\tuple {B, \diamond}$ be [[Definition:*-Algebra|$\ast$-algebras]] over $\C$.
Let $A \times B$ be the [[Definition:Direct Product of Algebras|direct product]] of $A$ and $B$.
Define $\ast : A \times B \to A \times B$ by:
:$\tuple {a, b}^\ast = \tuple {a^\square, b^\diamond}$
for each $\... | We show that $\ast$ is a [[Definition:Involution on Algebra|involution]]. | Direct Product of *-Algebras is *-Algebra | https://proofwiki.org/wiki/Direct_Product_of_*-Algebras_is_*-Algebra | https://proofwiki.org/wiki/Direct_Product_of_*-Algebras_is_*-Algebra | [
"*-Algebras"
] | [
"Definition:*-Algebra",
"Definition:Direct Product of Algebras",
"Definition:*-Algebra"
] | [
"Definition:Involution on Algebra"
] |
proofwiki-22168 | Direct Product of C*-Algebras is C*-Algebra | Let $\struct {A, \square, \norm {\, \cdot \,}_A}$ and $\struct {B, \diamond, \norm {\, \cdot \,}_B}$ be $\text C^\ast$-algebras.
Let $A \times B$ be the direct product of $A$ and $B$.
Define $\ast : A \times B \to A \times B$ by:
:$\tuple {a, b}^\ast = \tuple {a^\square, b^\diamond}$
for each $\tuple {a, b} \in A \time... | From Direct Product of Normed Algebras is Normed Algebra with Direct Product Norm, $\struct {A \times B, \norm {\, \cdot \,}_{A \times B} }$ is a normed algebra.
From Direct Product of Banach Spaces is Banach Space, $\struct {A \times B, \norm {\, \cdot \,}_{A \times B} }$ is a Banach algebra.
From Direct Product of ... | Let $\struct {A, \square, \norm {\, \cdot \,}_A}$ and $\struct {B, \diamond, \norm {\, \cdot \,}_B}$ be [[Definition:C*-Algebra|$\text C^\ast$-algebras]].
Let $A \times B$ be the [[Definition:Direct Product of Algebras|direct product]] of $A$ and $B$.
Define $\ast : A \times B \to A \times B$ by:
:$\tuple {a, b}^\ast... | From [[Direct Product of Normed Algebras is Normed Algebra with Direct Product Norm]], $\struct {A \times B, \norm {\, \cdot \,}_{A \times B} }$ is a [[Definition:Normed Algebra|normed algebra]].
From [[Direct Product of Banach Spaces is Banach Space]], $\struct {A \times B, \norm {\, \cdot \,}_{A \times B} }$ is a [... | Direct Product of C*-Algebras is C*-Algebra | https://proofwiki.org/wiki/Direct_Product_of_C*-Algebras_is_C*-Algebra | https://proofwiki.org/wiki/Direct_Product_of_C*-Algebras_is_C*-Algebra | [
"C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Direct Product of Algebras",
"Definition:Direct Product Norm",
"Definition:C*-Algebra"
] | [
"Direct Product of Normed Algebras is Normed Algebra with Direct Product Norm",
"Definition:Normed Algebra",
"Direct Product of Banach Spaces is Banach Space",
"Definition:Banach Algebra",
"Direct Product of *-Algebras is *-Algebra",
"Definition:*-Algebra",
"Category:C*-Algebras"
] |
proofwiki-22169 | Unitization of *-Algebra is *-Algebra | Let $\struct {A, \ast}$ be a $\ast$-algebra over $\C$.
Let $A_+$ be the unitization of $A$.
Define:
:$\tuple {a, \lambda}^\dagger = \tuple {a^\ast, \overline \lambda}$
for each $\tuple {a, \lambda} \in A_+$.
Then $\struct {A_+, \dagger}$ is a $\ast$-algebra. | === Proof of $(\text C^\ast 1)$ ===
Let $\tuple {a, \lambda} \in A_+$.
We have:
{{begin-eqn}}
{{eqn | l = \tuple {a, \lambda}^{\dagger \dagger }
| r = \tuple {a^\ast, \overline \lambda}^\dagger
}}
{{eqn | r = \tuple {a^{\ast \ast}, \overline {\overline \lambda} }
}}
{{eqn | r = \tuple {a, \lambda}
| c = $(\text C^\... | Let $\struct {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]] over $\C$.
Let $A_+$ be the [[Definition:Unitization of Algebra over Field|unitization]] of $A$.
Define:
:$\tuple {a, \lambda}^\dagger = \tuple {a^\ast, \overline \lambda}$
for each $\tuple {a, \lambda} \in A_+$.
Then $\struct {A_+, \dagger}$ is a... | === Proof of $(\text C^\ast 1)$ ===
Let $\tuple {a, \lambda} \in A_+$.
We have:
{{begin-eqn}}
{{eqn | l = \tuple {a, \lambda}^{\dagger \dagger }
| r = \tuple {a^\ast, \overline \lambda}^\dagger
}}
{{eqn | r = \tuple {a^{\ast \ast}, \overline {\overline \lambda} }
}}
{{eqn | r = \tuple {a, \lambda}
| c = $(\text C... | Unitization of *-Algebra is *-Algebra | https://proofwiki.org/wiki/Unitization_of_*-Algebra_is_*-Algebra | https://proofwiki.org/wiki/Unitization_of_*-Algebra_is_*-Algebra | [
"*-Algebras",
"Unitizations of Algebras over Fields"
] | [
"Definition:*-Algebra",
"Definition:Unitization of Algebra over Field",
"Definition:*-Algebra"
] | [] |
proofwiki-22170 | Continuous Functional Calculus of Reciprocal is Inverse Element | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $x \in A$ be normal and invertible.
Let $\map {\sigma_A} x$ be the spectrum of $x$ in $A$.
Let $\Theta_x$ be the continuous functional calculus for $x$.
Let $\iota : \map {\sigma_A} x \to \C$ be the inclusion.
Define $1/\iota : \map {... | Since $x$ is invertible, we have that $0 \not \in \map {\sigma_A} x$ and indeed $1/\iota$ is well-defined.
We have:
{{begin-eqn}}
{{eqn | l = x \map {\Theta_x} {1/\iota}
| r = \map {\Theta_x} \iota \map {\Theta_x} {1/\iota}
| c = {{Defof|Continuous Functional Calculus}}
}}
{{eqn | r = \map {\Theta_x} 1
| c = sinc... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $x \in A$ be [[Definition:Normal Element of *-Algebra|normal]] and [[Definition:Invertible Element|invertible]].
Let $\map {\sigma_A} x$ be the [[Definition:Spectrum (Sp... | Since $x$ is [[Definition:Invertible Element|invertible]], we have that $0 \not \in \map {\sigma_A} x$ and indeed $1/\iota$ is well-defined.
We have:
{{begin-eqn}}
{{eqn | l = x \map {\Theta_x} {1/\iota}
| r = \map {\Theta_x} \iota \map {\Theta_x} {1/\iota}
| c = {{Defof|Continuous Functional Calculus}}
}}
{{eqn |... | Continuous Functional Calculus of Reciprocal is Inverse Element | https://proofwiki.org/wiki/Continuous_Functional_Calculus_of_Reciprocal_is_Inverse_Element | https://proofwiki.org/wiki/Continuous_Functional_Calculus_of_Reciprocal_is_Inverse_Element | [
"Continuous Functional Calculus"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Normal Element of *-Algebra",
"Definition:Invertible Element",
"Definition:Spectrum (Spectral Theory)/Unital Algebra",
"Definition:Continuous Functional Calculus",
"Definition:Inclusion Mapping"
] | [
"Definition:Invertible Element",
"Definition:Algebra Homomorphism",
"Definition:Unital Algebra Homomorphism",
"Definition:Algebra Homomorphism",
"Definition:Unital Algebra Homomorphism",
"Category:Continuous Functional Calculus"
] |
proofwiki-22171 | Vector Subspace of Algebra over Field Embeds into Unitization as Vector Subspace | Let $K$ be a field.
Let $A$ be an algebra over $K$.
Let $B$ be a vector subspace of $A$.
Let $A_+$ be the unitization of $A$.
Let:
:$B_0 = \set {\tuple {x, 0_K} : x \in B}$
Then $B_0$ is a vector subspace of $A_+$. | Clearly $B_0 \ne \O$.
From One-Step Vector Subspace Test, it is sufficient to show that for each $u, v \in B_0$ and $\lambda \in K$, we have:
:$u + \lambda v \in B_0$
Let $u, v \in B_0$ and $\lambda \in K$.
Then there exists $x, y \in A$ such that:
:$u = \tuple {x, 0_K}$
and:
:$v = \tuple {y, 0_K}$
Then by the defini... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $A$ be an [[Definition:Algebra over Field|algebra over $K$]].
Let $B$ be a [[Definition:Vector Subspace|vector subspace]] of $A$.
Let $A_+$ be the [[Definition:Unitization of Algebra over Field|unitization]] of $A$.
Let:
:$B_0 = \set {\tuple {x, 0_K}... | Clearly $B_0 \ne \O$.
From [[One-Step Vector Subspace Test]], it is sufficient to show that for each $u, v \in B_0$ and $\lambda \in K$, we have:
:$u + \lambda v \in B_0$
Let $u, v \in B_0$ and $\lambda \in K$.
Then there exists $x, y \in A$ such that:
:$u = \tuple {x, 0_K}$
and:
:$v = \tuple {y, 0_K}$
Then by th... | Vector Subspace of Algebra over Field Embeds into Unitization as Vector Subspace | https://proofwiki.org/wiki/Vector_Subspace_of_Algebra_over_Field_Embeds_into_Unitization_as_Vector_Subspace | https://proofwiki.org/wiki/Vector_Subspace_of_Algebra_over_Field_Embeds_into_Unitization_as_Vector_Subspace | [
"Unitizations of Algebras over Fields"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Algebra over Field",
"Definition:Vector Subspace",
"Definition:Unitization of Algebra over Field",
"Definition:Vector Subspace"
] | [
"One-Step Vector Subspace Test",
"Definition:Unitization of Algebra over Field",
"One-Step Vector Subspace Test",
"Definition:Vector Subspace",
"Category:Unitizations of Algebras over Fields"
] |
proofwiki-22172 | Ideal of Algebra over Field Embeds into Unitization as Ideal | Let $K$ be a field.
Let $A$ be an algebra over $K$ that is not unital.
Let $I$ be an ideal of $A$.
Let $A_+$ be the unitization of $A$.
Let:
:$I_0 = \set {\tuple {x, 0_K} : x \in I}$
Then $I_0$ is an ideal of $A_+$. | From the definition of an ideal, $I$ is a vector subspace of $A$.
Hence from Vector Subspace of Algebra over Field Embeds into Unitization as Vector Subspace, $I_0$ is a vector subspace of $A_+$.
Let $\tuple {x, 0_K} \in I$ and $\tuple {y, \lambda} \in A_+$.
Then we have:
:$\tuple {x, 0_K} \tuple {y, \lambda} = \tup... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $A$ be an [[Definition:Algebra over Field|algebra over $K$]] that is not [[Definition:Unital Algebra|unital]].
Let $I$ be an [[Definition:Ideal of Algebra|ideal]] of $A$.
Let $A_+$ be the [[Definition:Unitization of Algebra over Field|unitization]] of ... | From the definition of an [[Definition:Ideal of Algebra|ideal]], $I$ is a [[Definition:Vector Subspace|vector subspace]] of $A$.
Hence from [[Vector Subspace of Algebra over Field Embeds into Unitization as Vector Subspace]], $I_0$ is a [[Definition:Vector Subspace|vector subspace]] of $A_+$.
Let $\tuple {x, 0_K} \... | Ideal of Algebra over Field Embeds into Unitization as Ideal | https://proofwiki.org/wiki/Ideal_of_Algebra_over_Field_Embeds_into_Unitization_as_Ideal | https://proofwiki.org/wiki/Ideal_of_Algebra_over_Field_Embeds_into_Unitization_as_Ideal | [
"Unitizations of Algebras over Fields"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Algebra over Field",
"Definition:Unital Algebra",
"Definition:Ideal of Algebra",
"Definition:Unitization of Algebra over Field",
"Definition:Ideal of Algebra"
] | [
"Definition:Ideal of Algebra",
"Definition:Vector Subspace",
"Vector Subspace of Algebra over Field Embeds into Unitization as Vector Subspace",
"Definition:Vector Subspace",
"Definition:Ideal of Algebra",
"Definition:Vector Subspace",
"Definition:Ideal of Algebra",
"Definition:Vector Subspace",
"Ca... |
proofwiki-22173 | Existence of Approximate Identity of C*-Algebra arising from Dense Ideal | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $I$ be an everywhere dense ideal of $A$.
Let $\le_A$ be the canonical preordering of $A$.
Let $A_{\mathbf{SA} }$ be the set of Hermitian elements of $A$.
Let:
:$\EE = \set {e \in I \cap A_{\mathbf{SA} } : \map {\sigma_A} e \subseteq \hointr... | Let $a, b \in \EE$.
We want to show that there exists $d \in \EE$ such that $a \le_A d$ and $b \le_A d$.
Let $A_+$ be the unitization of $A$.
Define:
:$\tuple {x, \lambda}^\ast = \tuple {x^\ast, \overline \lambda}$
for each $\tuple {x, \lambda}$.
From:
:Existence of Unitization of C*-Algebra if $A$ is non-unital
:Ex... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $I$ be an [[Definition:Everywhere Dense|everywhere dense]] [[Definition:Ideal of Algebra|ideal]] of $A$.
Let $\le_A$ be the [[Definition:Canonical Preordering of C*-Algebra|canonical preordering]] of $A$.
Let $A... | Let $a, b \in \EE$.
We want to show that there exists $d \in \EE$ such that $a \le_A d$ and $b \le_A d$.
Let $A_+$ be the [[Definition:Unitization of Normed Algebra|unitization]] of $A$.
Define:
:$\tuple {x, \lambda}^\ast = \tuple {x^\ast, \overline \lambda}$
for each $\tuple {x, \lambda}$.
From:
:[[Existence of... | Existence of Approximate Identity of C*-Algebra arising from Dense Ideal | https://proofwiki.org/wiki/Existence_of_Approximate_Identity_of_C*-Algebra_arising_from_Dense_Ideal | https://proofwiki.org/wiki/Existence_of_Approximate_Identity_of_C*-Algebra_arising_from_Dense_Ideal | [
"Approximate Identities of C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Everywhere Dense",
"Definition:Ideal of Algebra",
"Definition:Canonical Preordering of C*-Algebra",
"Definition:Set",
"Definition:Hermitian Element of *-Algebra",
"Definition:Directed Preordering",
"Definition:Approximate Identity of C*-Algebra"
] | [
"Definition:Unitization of Normed Algebra",
"Existence of Unitization of C*-Algebra",
"Definition:Unital Banach Algebra",
"Existence of Unique C* Norm on Unitization of Unital C*-Algebra",
"Definition:Unital Banach Algebra",
"Definition:Norm/Algebra",
"Definition:C*-Algebra",
"Definition:Identity (Abs... |
proofwiki-22174 | Set of Positive Elements of Everywhere Dense Ideal is Dense in Set of Positive Elements of C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $I$ be an everywhere dense ideal of $A$.
Let $A^+$ be the set of positive elements of $A$.
Let $I^+ = I \cap A^+$.
Then $I^+$ is everywhere dense in $A^+$. | Let $x \in A^+$.
From Existence and Uniqueness of Positive Nth Root of Positive Element of C*-Algebra, there exists a positive $y \in A$ such that $x = y^2$.
Since $I$ is everywhere dense in $A$, there exists a sequence $\sequence {y_n}_{n \mathop \in \N}$ in $I$ such that $y_n \to y$.
From Product of Element of C*-A... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $I$ be an [[Definition:Everywhere Dense|everywhere dense]] [[Definition:Ideal of Algebra|ideal]] of $A$.
Let $A^+$ be the [[Definition:Set|set]] of [[Definition:Positive Element of C*-Algebra|positive elements]] o... | Let $x \in A^+$.
From [[Existence and Uniqueness of Positive Nth Root of Positive Element of C*-Algebra]], there exists a [[Definition:Positive Element of C*-Algebra|positive]] $y \in A$ such that $x = y^2$.
Since $I$ is [[Definition:Everywhere Dense|everywhere dense]] in $A$, there exists a [[Definition:Sequence|se... | Set of Positive Elements of Everywhere Dense Ideal is Dense in Set of Positive Elements of C*-Algebra | https://proofwiki.org/wiki/Set_of_Positive_Elements_of_Everywhere_Dense_Ideal_is_Dense_in_Set_of_Positive_Elements_of_C*-Algebra | https://proofwiki.org/wiki/Set_of_Positive_Elements_of_Everywhere_Dense_Ideal_is_Dense_in_Set_of_Positive_Elements_of_C*-Algebra | [
"Positive Elements of C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Everywhere Dense",
"Definition:Ideal of Algebra",
"Definition:Set",
"Definition:Positive Element of C*-Algebra",
"Definition:Everywhere Dense"
] | [
"Existence and Uniqueness of Positive Nth Root of Positive Element of C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Everywhere Dense",
"Definition:Sequence",
"Product of Element of C*-Algebra with its Star is Positive",
"Definition:Positive Element of C*-Algebra",
"Definition:Ide... |
proofwiki-22175 | Approximate Identity for Everywhere Dense Subset is Approximate Identity of C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $\DD \subseteq A$ be everywhere dense.
Let $\tuple {\Lambda, \preceq}$ be a directed set.
Let $\family {e_\lambda}_{\lambda \in \Lambda}$ be a net such that:
:$e_\lambda$ is Hermitian for each $\lambda \in \Lambda$ with $\map {\sigma_A} {e... | We only need to show that for each $x \in A$, the net $\family {x e_\lambda}_{\lambda \mathop \in \Lambda}$ converges to $x$.
First, for each $\lambda \in \Lambda$ we have:
:$\map {\sigma_A} {e_\lambda} \subseteq \closedint 0 1$
Hence, from the definition of spectral radius:
:$\map {r_A} {e_\lambda} \le 1$
From Spect... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\DD \subseteq A$ be [[Definition:Everywhere Dense|everywhere dense]].
Let $\tuple {\Lambda, \preceq}$ be a [[Definition:Directed Set|directed set]].
Let $\family {e_\lambda}_{\lambda \in \Lambda}$ be a [[Defini... | We only need to show that for each $x \in A$, the [[Definition:Net (Preordered Set)|net]] $\family {x e_\lambda}_{\lambda \mathop \in \Lambda}$ [[Definition:Convergent Net|converges]] to $x$.
First, for each $\lambda \in \Lambda$ we have:
:$\map {\sigma_A} {e_\lambda} \subseteq \closedint 0 1$
Hence, from the defini... | Approximate Identity for Everywhere Dense Subset is Approximate Identity of C*-Algebra | https://proofwiki.org/wiki/Approximate_Identity_for_Everywhere_Dense_Subset_is_Approximate_Identity_of_C*-Algebra | https://proofwiki.org/wiki/Approximate_Identity_for_Everywhere_Dense_Subset_is_Approximate_Identity_of_C*-Algebra | [
"Approximate Identities of C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Everywhere Dense",
"Definition:Directed Preordering",
"Definition:Net (Preordered Set)",
"Definition:Hermitian Element of *-Algebra",
"Definition:Convergent Net",
"Definition:Approximate Identity of C*-Algebra"
] | [
"Definition:Net (Preordered Set)",
"Definition:Convergent Net",
"Definition:Spectral Radius",
"Spectral Radius of Normal Element of C*-Algebra Equal to Norm",
"Definition:Everywhere Dense",
"Definition:Sequence",
"Definition:Net (Preordered Set)",
"Definition:Convergent Net",
"Definition:Approximate... |
proofwiki-22176 | Locale Isomorphism is Isomorphism in Loc | Let $\mathbf{Loc}$ denote the category of locales.
Let $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ be locales.
Let $f:S_1 \to S_2$ be a continuous map of $\mathbf{Loc}$.
Then:
:$f$ is an isomorphism of $\mathbf{Loc}$ {{iff}} $f$ is a locale isomorphism. | By definition of category of locales:
:$\mathbf{Loc}$ is the dual category of the category of frames $\mathbf{Frm}$
From Isomorphism iff Isomorphism in Dual Category:
:$f$ is an isomorphism of $\mathbf{Loc}$
{{iff}}:
:$f = h^{\text{op}}$ for some isomorphism $h$ in $\mathbf{Frm}$
From Frame Isomorphism is Isomorphism i... | Let $\mathbf{Loc}$ denote the [[Definition:Category of Locales|category of locales]].
Let $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ be [[Definition:Locale (Lattice Theory)|locales]].
Let $f:S_1 \to S_2$ be a [[Definition:Continuous Map (Locale)|continuous map]] of $\mathbf{Loc}$.
Then:
:$... | By definition of [[Definition:Category of Locales|category of locales]]:
:$\mathbf{Loc}$ is the [[Definition:Dual Category|dual category]] of the [[Definition:Category of Frames|category of frames]] $\mathbf{Frm}$
From [[Isomorphism iff Isomorphism in Dual Category]]:
:$f$ is an [[Definition:Isomorphism (Category The... | Locale Isomorphism is Isomorphism in Loc | https://proofwiki.org/wiki/Locale_Isomorphism_is_Isomorphism_in_Loc | https://proofwiki.org/wiki/Locale_Isomorphism_is_Isomorphism_in_Loc | [
"Locales"
] | [
"Definition:Category of Locales",
"Definition:Locale (Lattice Theory)",
"Definition:Continuous Map (Locale)",
"Definition:Isomorphism",
"Definition:Locale Isomorphism"
] | [
"Definition:Category of Locales",
"Definition:Dual Category",
"Definition:Category of Frames",
"Isomorphism iff Isomorphism in Dual Category",
"Definition:Isomorphism (Category Theory)",
"Definition:Isomorphism (Category Theory)",
"Frame Isomorphism is Isomorphism in Category Frm",
"Definition:Isomorp... |
proofwiki-22177 | Star of Closed Subset of Banach *-Algebra is Closed | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a Banach $\ast$-algebra over $\C$.
Let $C \subseteq A$ be closed.
Let:
:$C^\ast = \set {x^\ast : x \in C}$
Then $C^\ast$ is closed. | Define $\phi : A \to A$ by $\map \phi a = a^\ast$ for each $a \in A$.
We have $C^\ast = \phi \sqbrk C$.
From the definition of a Banach $\ast$-algebra, we have $\norm {\map \phi a} = \norm a$.
We have $\map \phi {a - b} = \map \phi a - \map \phi b$ for each $a, b \in A$ so:
:$\norm {a - b} = \norm {\map \phi a - \map ... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Banach *-Algebra|Banach $\ast$-algebra]] over $\C$.
Let $C \subseteq A$ be [[Definition:Closed Set|closed]].
Let:
:$C^\ast = \set {x^\ast : x \in C}$
Then $C^\ast$ is [[Definition:Closed Set|closed]]. | Define $\phi : A \to A$ by $\map \phi a = a^\ast$ for each $a \in A$.
We have $C^\ast = \phi \sqbrk C$.
From the definition of a [[Definition:Banach *-Algebra|Banach $\ast$-algebra]], we have $\norm {\map \phi a} = \norm a$.
We have $\map \phi {a - b} = \map \phi a - \map \phi b$ for each $a, b \in A$ so:
:$\norm {... | Star of Closed Subset of Banach *-Algebra is Closed | https://proofwiki.org/wiki/Star_of_Closed_Subset_of_Banach_*-Algebra_is_Closed | https://proofwiki.org/wiki/Star_of_Closed_Subset_of_Banach_*-Algebra_is_Closed | [
"Banach *-Algebras"
] | [
"Definition:Banach *-Algebra",
"Definition:Closed Set",
"Definition:Closed Set"
] | [
"Definition:Banach *-Algebra",
"Image of Closed Set under Linear Isometry from Banach Space to Normed Vector Space is Closed",
"Definition:Closed Set",
"Category:Banach *-Algebras"
] |
proofwiki-22178 | Intersection of Algebra Ideals is Ideal | Let $K$ be a field.
Let $A$ be an algebra over $K$.
Let $\family {I_\alpha}_{\alpha \mathop \in I}$ be an $I$-indexed family of ideals of $A$.
Let:
:$\ds I = \bigcap_{\alpha \mathop \in I} I_\alpha$
Then $I$ is an ideal of $A$. | From Set of Linear Subspaces is Closed under Intersection, $I$ is a vector subspace of $A$.
Now, let $a, b \in I$.
Then we have $a, b \in I_\alpha$ for all $\alpha \in I$.
Since $I_\alpha$ is an ideal for each $\alpha \in I$, we have $a b \in I_\alpha$ and $b a \in I_\alpha$ for each $\alpha \in I$.
Hence we have $a b... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $A$ be an [[Definition:Algebra over Field|algebra]] over $K$.
Let $\family {I_\alpha}_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family of Sets|$I$-indexed family]] of [[Definition:Ideal of Algebra|ideals]] of $A$.
Let:
:$\ds I = \bigcap_{\alpha... | From [[Set of Linear Subspaces is Closed under Intersection]], $I$ is a [[Definition:Vector Subspace|vector subspace]] of $A$.
Now, let $a, b \in I$.
Then we have $a, b \in I_\alpha$ for all $\alpha \in I$.
Since $I_\alpha$ is an [[Definition:Ideal of Algebra|ideal]] for each $\alpha \in I$, we have $a b \in I_\alp... | Intersection of Algebra Ideals is Ideal | https://proofwiki.org/wiki/Intersection_of_Algebra_Ideals_is_Ideal | https://proofwiki.org/wiki/Intersection_of_Algebra_Ideals_is_Ideal | [
"Ideals of Algebras",
"Set Intersection"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Algebra over Field",
"Definition:Indexing Set/Family of Sets",
"Definition:Ideal of Algebra",
"Definition:Ideal of Algebra"
] | [
"Set of Linear Subspaces is Closed under Intersection",
"Definition:Vector Subspace",
"Definition:Ideal of Algebra",
"Definition:Ideal of Algebra",
"Category:Ideals of Algebras",
"Category:Set Intersection"
] |
proofwiki-22179 | Star of Vector Subspace of *-Algebra is Vector Subspace | Let $\struct {A, \ast}$ be a $\ast$-algebra over $\C$.
Let $B$ be a vector subspace of $A$.
Define:
:$B^\ast = \set {x^\ast : x \in B}$
Then $B^\ast$ is a vector subspace of $A$. | We use the One-Step Vector Subspace Test.
From Zero Vector in *-Algebra is Hermitian, we have:
:${\mathbf 0}_A^\ast = {\mathbf 0}_A$
Hence ${\mathbf 0}_A \in B$.
In particular $B \ne \O$.
Now let $u, v \in B^\ast$ and $\lambda \in \C$.
We can then write $u = x^\ast$ and $v = y^\ast$ for $x, y \in B$.
Then we have:
{{... | Let $\struct {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]] over $\C$.
Let $B$ be a [[Definition:Vector Subspace|vector subspace]] of $A$.
Define:
:$B^\ast = \set {x^\ast : x \in B}$
Then $B^\ast$ is a [[Definition:Vector Subspace|vector subspace]] of $A$. | We use the [[One-Step Vector Subspace Test]].
From [[Zero Vector in *-Algebra is Hermitian]], we have:
:${\mathbf 0}_A^\ast = {\mathbf 0}_A$
Hence ${\mathbf 0}_A \in B$.
In particular $B \ne \O$.
Now let $u, v \in B^\ast$ and $\lambda \in \C$.
We can then write $u = x^\ast$ and $v = y^\ast$ for $x, y \in B$.
Th... | Star of Vector Subspace of *-Algebra is Vector Subspace | https://proofwiki.org/wiki/Star_of_Vector_Subspace_of_*-Algebra_is_Vector_Subspace | https://proofwiki.org/wiki/Star_of_Vector_Subspace_of_*-Algebra_is_Vector_Subspace | [
"*-Algebras"
] | [
"Definition:*-Algebra",
"Definition:Vector Subspace",
"Definition:Vector Subspace"
] | [
"One-Step Vector Subspace Test",
"Zero Vector in *-Algebra is Hermitian",
"Definition:Vector Subspace",
"One-Step Vector Subspace Test",
"Definition:Vector Subspace",
"Category:*-Algebras"
] |
proofwiki-22180 | Star of Ideal of *-Algebra is Ideal | Let $\struct {A, \ast}$ be a $\ast$-algebra over $\C$.
Let $I$ be an ideal of $A$.
Define:
:$I^\ast = \set {x^\ast : x \in I}$
Then $I^\ast$ is an ideal of $A$. | From Star of Vector Subspace of *-Algebra is Vector Subspace, $I^\ast$ is a vector subspace of $A$.
Let $u \in I^\ast$ and $y \in A$.
Then $u = x^\ast$ for some $x \in I$.
Then we have:
:$u y = x^\ast y = \paren {y^\ast x}^\ast$
from $(\text C^\ast 1)$ and $(\text C^\ast 3)$ in the definition of an involution.
Since $x... | Let $\struct {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]] over $\C$.
Let $I$ be an [[Definition:Ideal of Algebra|ideal]] of $A$.
Define:
:$I^\ast = \set {x^\ast : x \in I}$
Then $I^\ast$ is an [[Definition:Ideal of Algebra|ideal]] of $A$. | From [[Star of Vector Subspace of *-Algebra is Vector Subspace]], $I^\ast$ is a [[Definition:Vector Subspace|vector subspace]] of $A$.
Let $u \in I^\ast$ and $y \in A$.
Then $u = x^\ast$ for some $x \in I$.
Then we have:
:$u y = x^\ast y = \paren {y^\ast x}^\ast$
from $(\text C^\ast 1)$ and $(\text C^\ast 3)$ in the... | Star of Ideal of *-Algebra is Ideal | https://proofwiki.org/wiki/Star_of_Ideal_of_*-Algebra_is_Ideal | https://proofwiki.org/wiki/Star_of_Ideal_of_*-Algebra_is_Ideal | [
"*-Algebras",
"Ideals of Algebras"
] | [
"Definition:*-Algebra",
"Definition:Ideal of Algebra",
"Definition:Ideal of Algebra"
] | [
"Star of Vector Subspace of *-Algebra is Vector Subspace",
"Definition:Vector Subspace",
"Definition:Involution on Algebra",
"Definition:Ideal of Algebra",
"Definition:Involution on Algebra",
"Definition:Ideal of Algebra",
"Category:*-Algebras",
"Category:Ideals of Algebras"
] |
proofwiki-22181 | Intersection of Ideal of *-Algebra with its Star is Self-Adjoint Ideal | Let $\struct {A, \ast}$ be a $\ast$-algebra over $\C$.
Let $I$ be an ideal of $A$.
Define:
:$I^\ast = \set {x^\ast : x \in I}$
Then $I \cap I^\ast$ is a self-adjoint ideal. | From Star of Ideal of *-Algebra is Ideal, $I^\ast$ is an ideal.
From Intersection of Algebra Ideals is Ideal, $I \cap I^\ast$ is an ideal.
Further, if $x \in I \cap I^\ast$, then $x \in I$ and there exists $y \in I$ such that $x = y^\ast$.
From $(\text C^\ast 1)$ in the definition of an involution, we have $x^\ast = y^... | Let $\struct {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]] over $\C$.
Let $I$ be an [[Definition:Ideal of Algebra|ideal]] of $A$.
Define:
:$I^\ast = \set {x^\ast : x \in I}$
Then $I \cap I^\ast$ is a [[Definition:Self-Adjoint Subset of *-Algebra|self-adjoint]] [[Definition:Ideal of Algebra|ideal]]. | From [[Star of Ideal of *-Algebra is Ideal]], $I^\ast$ is an [[Definition:Ideal of Algebra|ideal]].
From [[Intersection of Algebra Ideals is Ideal]], $I \cap I^\ast$ is an [[Definition:Ideal of Algebra|ideal]].
Further, if $x \in I \cap I^\ast$, then $x \in I$ and there exists $y \in I$ such that $x = y^\ast$.
From ... | Intersection of Ideal of *-Algebra with its Star is Self-Adjoint Ideal | https://proofwiki.org/wiki/Intersection_of_Ideal_of_*-Algebra_with_its_Star_is_Self-Adjoint_Ideal | https://proofwiki.org/wiki/Intersection_of_Ideal_of_*-Algebra_with_its_Star_is_Self-Adjoint_Ideal | [
"*-Algebras",
"Ideals of Algebras"
] | [
"Definition:*-Algebra",
"Definition:Ideal of Algebra",
"Definition:Self-Adjoint Subset of *-Algebra",
"Definition:Ideal of Algebra"
] | [
"Star of Ideal of *-Algebra is Ideal",
"Definition:Ideal of Algebra",
"Intersection of Algebra Ideals is Ideal",
"Definition:Ideal of Algebra",
"Definition:Involution on Algebra",
"Definition:Self-Adjoint Subset of *-Algebra",
"Category:*-Algebras",
"Category:Ideals of Algebras"
] |
proofwiki-22182 | Norm of Element of Approximate Identity of C*-Algebra is Less Than or equal to One | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $\family {e_\lambda}_{\lambda \in \Lambda}$ be an approximate identity of $A$.
Then for each $\lambda \in \Lambda$ we have:
:$\norm {e_\lambda} \le 1$ | From the definition of an approximate identity, for each $\lambda \in \Lambda$ we have:
:$\map {\sigma_A} {e_\lambda} \subseteq \closedint 0 1$
So that, from the definition of spectral radius:
:$\map {r_A} {e_\lambda} \le 1$
By definition, each $e_\lambda$ is positive and hence Hermitian.
Hence from Spectral Radius of... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\family {e_\lambda}_{\lambda \in \Lambda}$ be an [[Definition:Approximate Identity of C*-Algebra|approximate identity]] of $A$.
Then for each $\lambda \in \Lambda$ we have:
:$\norm {e_\lambda} \le 1$ | From the definition of an [[Definition:Approximate Identity of C*-Algebra|approximate identity]], for each $\lambda \in \Lambda$ we have:
:$\map {\sigma_A} {e_\lambda} \subseteq \closedint 0 1$
So that, from the definition of [[Definition:Spectral Radius|spectral radius]]:
:$\map {r_A} {e_\lambda} \le 1$
By definiti... | Norm of Element of Approximate Identity of C*-Algebra is Less Than or equal to One | https://proofwiki.org/wiki/Norm_of_Element_of_Approximate_Identity_of_C*-Algebra_is_Less_Than_or_equal_to_One | https://proofwiki.org/wiki/Norm_of_Element_of_Approximate_Identity_of_C*-Algebra_is_Less_Than_or_equal_to_One | [
"Approximate Identities of C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Approximate Identity of C*-Algebra"
] | [
"Definition:Approximate Identity of C*-Algebra",
"Definition:Spectral Radius",
"Definition:Positive Element of C*-Algebra",
"Definition:Hermitian Element of *-Algebra",
"Spectral Radius of Normal Element of C*-Algebra Equal to Norm",
"Category:Approximate Identities of C*-Algebras"
] |
proofwiki-22183 | Closed Ideal of C*-Algebra is Self-Adjoint | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $I$ be a closed ideal of $A$.
Then for each $x \in I$, we have $x^\ast \in I$.
That is, $I$ is self-adjoint. | Let $x \in I$.
Since $I$ is an ideal, we have $x^\ast x \in I$.
From Generalized Polar Decomposition in C*-Algebra, there exists $u \in A$ such that $x = u \paren {\paren {x^\ast x}^{1/2} }^{1/2}$.
From Power of Power of Positive Element of Unital C*-Algebra, we have $x = u \paren {x^\ast x}^{1/4}$.
From Continuous F... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $I$ be a [[Definition:Closed Set|closed]] [[Definition:Ideal of Algebra|ideal]] of $A$.
Then for each $x \in I$, we have $x^\ast \in I$.
That is, $I$ is [[Definition:Self-Adjoint Subset of *-Algebra|self-adjoin... | Let $x \in I$.
Since $I$ is an [[Definition:Ideal of Algebra|ideal]], we have $x^\ast x \in I$.
From [[Generalized Polar Decomposition in C*-Algebra]], there exists $u \in A$ such that $x = u \paren {\paren {x^\ast x}^{1/2} }^{1/2}$.
From [[Power of Power of Positive Element of Unital C*-Algebra]], we have $x = u \... | Closed Ideal of C*-Algebra is Self-Adjoint/Proof 2 | https://proofwiki.org/wiki/Closed_Ideal_of_C*-Algebra_is_Self-Adjoint | https://proofwiki.org/wiki/Closed_Ideal_of_C*-Algebra_is_Self-Adjoint/Proof_2 | [
"Closed Ideal of C*-Algebra is Self-Adjoint",
"C*-Algebras",
"Closed Ideal of C*-Algebra is Self-Adjoint"
] | [
"Definition:C*-Algebra",
"Definition:Closed Set",
"Definition:Ideal of Algebra",
"Definition:Self-Adjoint Subset of *-Algebra"
] | [
"Definition:Ideal of Algebra",
"Generalized Polar Decomposition in C*-Algebra",
"Power of Power of Positive Element of Unital C*-Algebra",
"Continuous Function applied to Normal Element of Unital C*-Algebra is Positive iff Function is Non-Negative",
"Definition:Positive Element of C*-Algebra",
"Definition... |
proofwiki-22184 | Mapping is Bijection iff Direct Image Mapping is Bijection | Let $f: S \to T$ be a mapping.
Let $f^\to: \powerset S \to \powerset T$ be the direct image mapping of $f$.
Then:
:$f^\to$ is a bijection
{{iff}}
:$f: S \to T$ is also a bijection. | Follows immediately from:
* Mapping is Injection iff Direct Image Mapping is Injection
* Mapping is Surjection iff Direct Image Mapping is Surjection
{{qed}}
Category:Bijections
Category:Direct Image Mappings
3wluh86o3werc9punas37r6q77z8pth | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $f^\to: \powerset S \to \powerset T$ be the [[Definition:Direct Image Mapping of Mapping|direct image mapping]] of $f$.
Then:
:$f^\to$ is a [[Definition:Bijection|bijection]]
{{iff}}
:$f: S \to T$ is also a [[Definition:Bijection|bijection]]. | Follows immediately from:
* [[Mapping is Injection iff Direct Image Mapping is Injection]]
* [[Mapping is Surjection iff Direct Image Mapping is Surjection]]
{{qed}}
[[Category:Bijections]]
[[Category:Direct Image Mappings]]
3wluh86o3werc9punas37r6q77z8pth | Mapping is Bijection iff Direct Image Mapping is Bijection | https://proofwiki.org/wiki/Mapping_is_Bijection_iff_Direct_Image_Mapping_is_Bijection | https://proofwiki.org/wiki/Mapping_is_Bijection_iff_Direct_Image_Mapping_is_Bijection | [
"Bijections",
"Direct Image Mappings"
] | [
"Definition:Mapping",
"Definition:Direct Image Mapping/Mapping",
"Definition:Bijection",
"Definition:Bijection"
] | [
"Mapping is Injection iff Direct Image Mapping is Injection",
"Mapping is Surjection iff Direct Image Mapping is Surjection",
"Category:Bijections",
"Category:Direct Image Mappings"
] |
proofwiki-22185 | Figures in Perspective from Point are in Perspective from Line | Let $A$ and $B$ be plane geometric figures.
Let $A$ and $B$ be in perspective from a point.
Then $A$ and $B$ are also in perspective from a line. | {{ProofWanted|A consequence of Desargues' Theorem}} | Let $A$ and $B$ be [[Definition:Plane Figure|plane geometric figures]].
Let $A$ and $B$ be in [[Definition:Perspective from Point|perspective from a point]].
Then $A$ and $B$ are also in [[Definition:Perspective from Line|perspective from a line]]. | {{ProofWanted|A consequence of [[Desargues' Theorem]]}} | Figures in Perspective from Point are in Perspective from Line | https://proofwiki.org/wiki/Figures_in_Perspective_from_Point_are_in_Perspective_from_Line | https://proofwiki.org/wiki/Figures_in_Perspective_from_Point_are_in_Perspective_from_Line | [
"Perspective"
] | [
"Definition:Geometric Figure/Plane Figure",
"Definition:Perspective/Point",
"Definition:Perspective/Line"
] | [
"Desargues' Theorem"
] |
proofwiki-22186 | Figures in Perspective from Line are in Perspective from Point | Let $A$ and $B$ be plane geometric figures.
Let $A$ and $B$ be in perspective from a line.
Then $A$ and $B$ are also in perspective from a point. | {{ProofWanted|A consequence of Desargues' Theorem}} | Let $A$ and $B$ be [[Definition:Plane Figure|plane geometric figures]].
Let $A$ and $B$ be in [[Definition:Perspective from Line|perspective from a line]].
Then $A$ and $B$ are also in [[Definition:Perspective from Point|perspective from a point]]. | {{ProofWanted|A consequence of [[Desargues' Theorem]]}} | Figures in Perspective from Line are in Perspective from Point | https://proofwiki.org/wiki/Figures_in_Perspective_from_Line_are_in_Perspective_from_Point | https://proofwiki.org/wiki/Figures_in_Perspective_from_Line_are_in_Perspective_from_Point | [
"Perspective"
] | [
"Definition:Geometric Figure/Plane Figure",
"Definition:Perspective/Line",
"Definition:Perspective/Point"
] | [
"Desargues' Theorem"
] |
proofwiki-22187 | Separable C*-Algebra has Sequential Approximate Identity | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a separable $\text C^\ast$ algebra.
Then there exists a sequence $\family {e_n}_{n \mathop \in \N}$ forming an approximate identity of $A$. | Let $\set {a_n : n \in \N}$ be a countable everywhere dense subset of $A$.
From Existence of Approximate Identity of C*-Algebra arising from Dense Ideal, there exists a directed set $\struct {\Lambda, \preceq}$ and net $\family {e_\lambda}_{\lambda \mathop \in \Lambda}$ forming an approximate identity of $A$.
The issue... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Separable Space|separable]] [[Definition:C*-Algebra|$\text C^\ast$ algebra]].
Then there exists a [[Definition:Sequence|sequence]] $\family {e_n}_{n \mathop \in \N}$ forming an [[Definition:Approximate Identity of C*-Algebra|approximate identity]] of $A$. | Let $\set {a_n : n \in \N}$ be a [[Definition:Countable Set|countable]] [[Definition:Everywhere Dense|everywhere dense subset]] of $A$.
From [[Existence of Approximate Identity of C*-Algebra arising from Dense Ideal]], there exists a [[Definition:Directed Set|directed set]] $\struct {\Lambda, \preceq}$ and [[Definitio... | Separable C*-Algebra has Sequential Approximate Identity | https://proofwiki.org/wiki/Separable_C*-Algebra_has_Sequential_Approximate_Identity | https://proofwiki.org/wiki/Separable_C*-Algebra_has_Sequential_Approximate_Identity | [
"Approximate Identities of C*-Algebras"
] | [
"Definition:Separable Space",
"Definition:C*-Algebra",
"Definition:Sequence",
"Definition:Approximate Identity of C*-Algebra"
] | [
"Definition:Countable Set",
"Definition:Everywhere Dense",
"Existence of Approximate Identity of C*-Algebra arising from Dense Ideal",
"Definition:Directed Preordering",
"Definition:Net (Preordered Set)",
"Definition:Approximate Identity of C*-Algebra",
"Definition:Countable Set",
"Definition:Increasi... |
proofwiki-22188 | Spectrum of Locale as Completely Prime Filters is Sober Space | Let $\struct{L, \preceq}$ be a locale.
Let $\map {\operatorname{Sp}} L$ denote the spectrum as completely prime filters of $L$.
Then:
:$\map {\operatorname{Sp}} L$ is a sober space. | By definition of spectrum as completely prime filters:
:$\map {\operatorname{Sp}} L = \struct{\map {\operatorname{pt}} L, \set{\Sigma_a : a \in L}}$
where:
:$\map {\operatorname{pt}} L$ denotes the set of points as completely prime filters of $L$
:$\forall a \in L : \Sigma_a = \set{p \in \map {\operatorname{pt}} L : a ... | Let $\struct{L, \preceq}$ be a [[Definition:Locale (Lattice Theory)|locale]].
Let $\map {\operatorname{Sp}} L$ denote the [[Definition:Spectrum of Locale as Completely Prime Filters|spectrum as completely prime filters]] of $L$.
Then:
:$\map {\operatorname{Sp}} L$ is a [[Definition:Sober Space|sober space]]. | By definition of [[Definition:Spectrum of Locale as Completely Prime Filters|spectrum as completely prime filters]]:
:$\map {\operatorname{Sp}} L = \struct{\map {\operatorname{pt}} L, \set{\Sigma_a : a \in L}}$
where:
:$\map {\operatorname{pt}} L$ denotes the [[Definition:Set|set]] of [[Definition:Point of Locale as Co... | Spectrum of Locale as Completely Prime Filters is Sober Space | https://proofwiki.org/wiki/Spectrum_of_Locale_as_Completely_Prime_Filters_is_Sober_Space | https://proofwiki.org/wiki/Spectrum_of_Locale_as_Completely_Prime_Filters_is_Sober_Space | [
"Spectra of Locales"
] | [
"Definition:Locale (Lattice Theory)",
"Definition:Spectrum of Locale/Completely Prime Filters",
"Definition:Sober Space"
] | [
"Definition:Spectrum of Locale/Completely Prime Filters",
"Definition:Set",
"Definition:Point of Locale/Completely Prime Filter",
"Definition:Mapping",
"Definition:Power Set",
"Canonical Mapping of Locale to Powerset of Points is Frame Homomorphism",
"Definition:Frame Homomorphism",
"Definition:Frame ... |
proofwiki-22189 | Norm of Non-Negative Increasing Continuous Function applied to Positive Element of Unital C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $x \in A$ be positive.
Let $\map {\sigma_A} x \subseteq \hointr 0 \infty$ be the spectrum of $x$ in $A$.
Let $f : \map {\sigma_A} x \to \hointr 0 \infty$ be an increasing continuous function.
Let $\map f x$ be obtained from the contin... | From Continuous Function applied to Normal Element of Unital C*-Algebra is Positive iff Function is Non-Negative:
:$\map f x$ is positive.
We then have:
{{begin-eqn}}
{{eqn | l = \norm {\map f x}
| r = \sup \set {\cmod \mu : \mu \in \map {\sigma_A} {\map f x} }
| c = Spectral Radius of Normal Element of C*-Algebra ... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $x \in A$ be [[Definition:Positive Element of C*-Algebra|positive]].
Let $\map {\sigma_A} x \subseteq \hointr 0 \infty$ be the [[Definition:Spectrum (Spectral Theory)|sp... | From [[Continuous Function applied to Normal Element of Unital C*-Algebra is Positive iff Function is Non-Negative]]:
:$\map f x$ is [[Definition:Positive Element of C*-Algebra|positive]].
We then have:
{{begin-eqn}}
{{eqn | l = \norm {\map f x}
| r = \sup \set {\cmod \mu : \mu \in \map {\sigma_A} {\map f x} }
| c... | Norm of Non-Negative Increasing Continuous Function applied to Positive Element of Unital C*-Algebra | https://proofwiki.org/wiki/Norm_of_Non-Negative_Increasing_Continuous_Function_applied_to_Positive_Element_of_Unital_C*-Algebra | https://proofwiki.org/wiki/Norm_of_Non-Negative_Increasing_Continuous_Function_applied_to_Positive_Element_of_Unital_C*-Algebra | [
"Positive Elements of C*-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Spectrum (Spectral Theory)",
"Definition:Increasing/Real Function",
"Definition:Continuous Function",
"Definition:Continuous Functional Calculus"
] | [
"Continuous Function applied to Normal Element of Unital C*-Algebra is Positive iff Function is Non-Negative",
"Definition:Positive Element of C*-Algebra",
"Spectral Radius of Normal Element of C*-Algebra Equal to Norm",
"Spectral Mapping Theorem",
"Definition:Increasing/Real Function",
"Spectral Radius o... |
proofwiki-22190 | Homeomorphism Preserves Sobriety | Let $T_1 = \struct{S_1, \tau_1}$ be a sober space.
Let $T_2 = \struct{S_2, \tau_2}$ be a topological space.
Let $f : T_1 \to T_2$ be a homeomorphism.
Then:
:$T_2$ is a sober space. | From Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods it is sufficent to show:
:for each completely prime filter $\FF$ in the complete lattice $\struct{\tau_2, \subseteq}$:
::$\exists ! y \in S_2 : \FF = \map \UU y$
Let $f^{-1}$ denote the inverse of $f$.
From Inverse of Homeomorphism is H... | Let $T_1 = \struct{S_1, \tau_1}$ be a [[Definition:Sober Space|sober space]].
Let $T_2 = \struct{S_2, \tau_2}$ be a [[Definition:Topological Space|topological space]].
Let $f : T_1 \to T_2$ be a [[Definition:Homeomorphism (Topological Spaces)|homeomorphism]].
Then:
:$T_2$ is a [[Definition:Sober Space|sober space]... | From [[Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods]] it is sufficent to show:
:for each [[Definition:Completely Prime Filter|completely prime filter]] $\FF$ in the [[Definition:Complete Lattice|complete lattice]] $\struct{\tau_2, \subseteq}$:
::$\exists ! y \in S_2 : \FF = \map \UU y$... | Homeomorphism Preserves Sobriety | https://proofwiki.org/wiki/Homeomorphism_Preserves_Sobriety | https://proofwiki.org/wiki/Homeomorphism_Preserves_Sobriety | [
"Sober Spaces",
"Homeomorphisms (Topological Spaces)"
] | [
"Definition:Sober Space",
"Definition:Topological Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Sober Space"
] | [
"Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods",
"Definition:Completely Prime Filter",
"Definition:Complete Lattice",
"Definition:Inverse of Mapping",
"Inverse of Homeomorphism is Homeomorphism",
"Definition:Homeomorphism",
"Definition:Completely Prime Filter",
"Frame ... |
proofwiki-22191 | Factorization Theorem for C*-Algebra in terms of Bound on Modulus | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $\le_A$ be the canonical preordering of $A$.
Let $a \in A$ be positive.
Let $x \in A$ be such that:
:$x^\ast x \le_A a$
That is:
:$\cmod x^2 \le_A a$
where $\cmod x$ is the modulus of $x$.
Let $0 < \alpha < 1/2$.
Then there exists $u \in A... | First take $A$ unital. | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\le_A$ be the [[Definition:Canonical Preordering of C*-Algebra|canonical preordering]] of $A$.
Let $a \in A$ be [[Definition:Positive Element of C*-Algebra|positive]].
Let $x \in A$ be such that:
:$x^\ast x \le... | First take $A$ [[Definition:Unital Banach Algebra|unital]]. | Factorization Theorem for C*-Algebra in terms of Bound on Modulus | https://proofwiki.org/wiki/Factorization_Theorem_for_C*-Algebra_in_terms_of_Bound_on_Modulus | https://proofwiki.org/wiki/Factorization_Theorem_for_C*-Algebra_in_terms_of_Bound_on_Modulus | [
"Factorization Lemma for C*-Algebra in terms of Bound on Modulus",
"Factorization Theorem for C*-Algebra in terms of Bound on Modulus",
"Factorization Theorem for C*-Algebra in terms of Bound on Modulus",
"C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Canonical Preordering of C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Modulus of Element of C*-Algebra",
"Definition:Unital Banach Algebra",
"Definition:Continuous Functional Calculus"
] | [
"Definition:Unital Banach Algebra",
"Definition:Unital Banach Algebra",
"Definition:Unital Banach Algebra",
"Definition:Unital Banach Algebra",
"Definition:Unital Banach Algebra"
] |
proofwiki-22192 | Product of Powers of Positive Element of Unital C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $a \in A$ be positive elements.
Let $\alpha, \beta > 0$ be real numbers.
Let $a^\alpha$, $a^\beta$ and $a^{\alpha + \beta}$ be given by the continuous functional calculus.
Then:
:$a^{\alpha + \beta} = a^\alpha a^\beta$ | Let $\map {\sigma_A} a \subseteq \hointr 0 \infty$ be the spectrum of $a$.
Define $e_\alpha : \map {\sigma_A} a \to \hointr 0 \infty$, $e_\beta : \map {\sigma_A} a \to \hointr 0 \infty$ and $e_{\alpha + \beta} : \map {\sigma_A} a \to \hointr 0 \infty$ by:
:$\map {e_\alpha} t = t^\alpha$
:$\map {e_\beta} t = t^\beta$
:$... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $a \in A$ be [[Definition:Positive Element of C*-Algebra|positive elements]].
Let $\alpha, \beta > 0$ be [[Definition:Real Number|real numbers]].
Let $a^\alpha$, $a^\be... | Let $\map {\sigma_A} a \subseteq \hointr 0 \infty$ be the [[Definition:Spectrum (Spectral Theory)/Non-Unital Algebra|spectrum]] of $a$.
Define $e_\alpha : \map {\sigma_A} a \to \hointr 0 \infty$, $e_\beta : \map {\sigma_A} a \to \hointr 0 \infty$ and $e_{\alpha + \beta} : \map {\sigma_A} a \to \hointr 0 \infty$ by:
:$... | Product of Powers of Positive Element of Unital C*-Algebra | https://proofwiki.org/wiki/Product_of_Powers_of_Positive_Element_of_Unital_C*-Algebra | https://proofwiki.org/wiki/Product_of_Powers_of_Positive_Element_of_Unital_C*-Algebra | [
"C*-Algebras",
"Positive Elements of C*-Algebras",
"Positive Elements of C*-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Real Number",
"Definition:Continuous Functional Calculus"
] | [
"Definition:Spectrum (Spectral Theory)/Non-Unital Algebra",
"Exponent Combination Laws/Product of Powers",
"Definition:Continuous Functional Calculus",
"Definition:Algebra Homomorphism",
"Category:Positive Elements of C*-Algebras"
] |
proofwiki-22193 | Power of Power of Positive Element of Unital C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $a \in A$ be positive elements.
Let $\alpha, \beta > 0$ be real numbers.
Let $a^{\alpha \beta}$ be given by the continuous functional calculus for $a$.
Let $\paren {a^\alpha}^\beta$ be given by the continuous functional calculus for $a^\alp... | First take $A$ unital.
Let $\map {\sigma_A} a$ be the spectrum of $a$ in $A$.
Let $\map {\sigma_A} {a^\alpha}$ be the spectrum of $a^\alpha$ in $A$.
From Continuous Function applied to Normal Element of Unital C*-Algebra is Positive iff Function is Non-Negative, $a^\alpha$ is positive.
Let $\Theta_{a^\alpha}$ be the co... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $a \in A$ be [[Definition:Positive Element of C*-Algebra|positive elements]].
Let $\alpha, \beta > 0$ be [[Definition:Real Number|real numbers]].
Let $a^{\alpha \beta}$ be given by the [[Definition:Continuous Fu... | First take $A$ [[Definition:Unital Banach Algebra|unital]].
Let $\map {\sigma_A} a$ be the [[Definition:Spectrum (Spectral Theory)/Unital Algebra|spectrum]] of $a$ in $A$.
Let $\map {\sigma_A} {a^\alpha}$ be the [[Definition:Spectrum (Spectral Theory)/Unital Algebra|spectrum]] of $a^\alpha$ in $A$.
From [[Continuous... | Power of Power of Positive Element of Unital C*-Algebra | https://proofwiki.org/wiki/Power_of_Power_of_Positive_Element_of_Unital_C*-Algebra | https://proofwiki.org/wiki/Power_of_Power_of_Positive_Element_of_Unital_C*-Algebra | [
"Positive Elements of C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Real Number",
"Definition:Continuous Functional Calculus",
"Definition:Continuous Functional Calculus"
] | [
"Definition:Unital Banach Algebra",
"Definition:Spectrum (Spectral Theory)/Unital Algebra",
"Definition:Spectrum (Spectral Theory)/Unital Algebra",
"Continuous Function applied to Normal Element of Unital C*-Algebra is Positive iff Function is Non-Negative",
"Definition:Positive Element of C*-Algebra",
"D... |
proofwiki-22194 | Real-Valued Continuous Function Vanishing at Zero applied to Hermitian Element of Closed Ideal of Unital C*-Algebra is contained in Ideal | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $I$ be a closed ideal.
Let $x \in I$ be Hermitian.
Let $\map {\sigma_A} x \subseteq \R$ be the spectrum of $x$ in $A$.
Let $K$ be a compact set such that $0 \in K$ and $\map {\sigma_A} x \subseteq K$.
Let $f : K \to \R$ be a continu... | Let $\norm {\, \cdot \,}_\infty$ be the supremum norm on $\map \CC {K, \R}$.
From the Weierstrass Approximation Theorem, there exists a sequence of polynomials $\sequence {p_n}_{n \mathop \in \N}$ such that:
:$\norm {p_n - f}_\infty \to 0$ as $n \to \infty$.
We have:
:$\map {p_n} 0 \to \map f 0 = 0$ as $n \to \infty$.... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $I$ be a [[Definition:Closed Set|closed]] [[Definition:Ideal of Algebra|ideal]].
Let $x \in I$ be [[Definition:Hermitian Element of *-Algebra|Hermitian]].
Let $\map {\s... | Let $\norm {\, \cdot \,}_\infty$ be the [[Definition:Supremum Norm|supremum norm]] on $\map \CC {K, \R}$.
From the [[Weierstrass Approximation Theorem]], there exists a [[Definition:Sequence|sequence]] of [[Definition:Polynomial|polynomials]] $\sequence {p_n}_{n \mathop \in \N}$ such that:
:$\norm {p_n - f}_\infty \t... | Real-Valued Continuous Function Vanishing at Zero applied to Hermitian Element of Closed Ideal of Unital C*-Algebra is contained in Ideal | https://proofwiki.org/wiki/Real-Valued_Continuous_Function_Vanishing_at_Zero_applied_to_Hermitian_Element_of_Closed_Ideal_of_Unital_C*-Algebra_is_contained_in_Ideal | https://proofwiki.org/wiki/Real-Valued_Continuous_Function_Vanishing_at_Zero_applied_to_Hermitian_Element_of_Closed_Ideal_of_Unital_C*-Algebra_is_contained_in_Ideal | [
"Continuous Functional Calculus",
"Real-Valued Continuous Function Vanishing at Zero applied to Hermitian Element of Closed Ideal of Unital C*-Algebra is contained in Ideal"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Closed Set",
"Definition:Ideal of Algebra",
"Definition:Hermitian Element of *-Algebra",
"Definition:Spectrum (Spectral Theory)/Unital Algebra",
"Definition:Compact Topological Space",
"Definition:Continuous Function",
"Definit... | [
"Definition:Supremum Norm",
"Weierstrass Approximation Theorem",
"Definition:Sequence",
"Definition:Polynomial",
"Definition:Ideal of Algebra",
"Definition:Vector Subspace",
"Definition:Continuous Functional Calculus",
"Definition:Linear Isometry",
"Definition:Closed Set",
"Category:Continuous Fun... |
proofwiki-22195 | Equation of Plane/General Equation | A plane $P$ is the set of all $\tuple {x, y, z} \in \R^3$, where:
:$\alpha_1 x + \alpha_2 y + \alpha_3 z = \gamma$
where $\alpha_1, \alpha_2, \alpha_3, \gamma \in \R$ are given, and not all of $\alpha_1, \alpha_2, \alpha_3$ are zero. | Let $\bsalpha$ be a non-zero vector that is perpendicular to $P$.
Let $A$ be a point in $P$ such that $A = k \bsalpha$ for some $k \in \mathbb R$.
Let $X$ be any other point in $P$.
Let $\mathbf x$ be the vector from the origin to $X$.
Thus, the vector to $X$ from $A$ is $\mathbf x - k \bsalpha$.
By the definition of a... | A [[Definition:Plane|plane]] $P$ is the [[Definition:Set|set]] of all $\tuple {x, y, z} \in \R^3$, where:
:$\alpha_1 x + \alpha_2 y + \alpha_3 z = \gamma$
where $\alpha_1, \alpha_2, \alpha_3, \gamma \in \R$ are given, and not all of $\alpha_1, \alpha_2, \alpha_3$ are [[Definition:Zero (Number)|zero]]. | Let $\bsalpha$ be a non-[[Definition:Zero Vector|zero]] [[Definition:Position Vector|vector]] that is [[Definition:Perpendicular|perpendicular]] to $P$.
Let $A$ be a [[Definition:Point|point]] in $P$ such that $A = k \bsalpha$ for some $k \in \mathbb R$.
Let $X$ be any other [[Definition:Point|point]] in $P$.
Let $\... | Equation of Plane/General Equation | https://proofwiki.org/wiki/Equation_of_Plane/General_Equation | https://proofwiki.org/wiki/Equation_of_Plane/General_Equation | [
"General Equation of Plane",
"Equations of Planes"
] | [
"Definition:Plane Surface",
"Definition:Set",
"Definition:Zero (Number)"
] | [
"Definition:Zero Vector",
"Definition:Position Vector",
"Definition:Right Angle/Perpendicular",
"Definition:Point",
"Definition:Point",
"Definition:Position Vector",
"Definition:Coordinate System/Origin",
"Definition:Position Vector",
"Definition:Right Angle/Perpendicular/Plane",
"Definition:Right... |
proofwiki-22196 | Factorization Theorem for C*-Algebra in terms of Bound on Modulus/Lemma | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $\le_A$ be the canonical preordering of $A$.
Let $\alpha, \beta \in \R$ be such that $\alpha + \beta > 1$.
Let $a \in A$ be positive.
Let $x, y \in A$ be such that:
:$x^\ast x \le_A a^\alpha$
and:
:$y y^\ast \le_A a^\beta$
Let:
:$u_... | Define:
:$d_{n, m} = \paren {n^{-1} {\mathbf 1}_A + a}^{-1/2} - \paren {m^{-1} {\mathbf 1}_A + a}^{-1/2}$
From Product of Element of C*-Algebra with its Star is Positive, we have:
:${\mathbf 0}_A \le_A x^\ast x \le_A a^\alpha$
and:
:${\mathbf 0}_A \le_A y^\ast y \le_A a^\beta$
From Continuous Function applied to Norma... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\le_A$ be the [[Definition:Canonical Preordering of C*-Algebra|canonical preordering]] of $A$.
Let $\alpha, \beta \in \R$ be such that $\alpha + \beta > 1$.
Let $a \... | Define:
:$d_{n, m} = \paren {n^{-1} {\mathbf 1}_A + a}^{-1/2} - \paren {m^{-1} {\mathbf 1}_A + a}^{-1/2}$
From [[Product of Element of C*-Algebra with its Star is Positive]], we have:
:${\mathbf 0}_A \le_A x^\ast x \le_A a^\alpha$
and:
:${\mathbf 0}_A \le_A y^\ast y \le_A a^\beta$
From [[Continuous Function applied ... | Factorization Theorem for C*-Algebra in terms of Bound on Modulus/Lemma | https://proofwiki.org/wiki/Factorization_Theorem_for_C*-Algebra_in_terms_of_Bound_on_Modulus/Lemma | https://proofwiki.org/wiki/Factorization_Theorem_for_C*-Algebra_in_terms_of_Bound_on_Modulus/Lemma | [
"Factorization Theorem for C*-Algebra in terms of Bound on Modulus"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Canonical Preordering of C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Continuous Functional Calculus",
"Definition:Convergent Sequence"
] | [
"Product of Element of C*-Algebra with its Star is Positive",
"Continuous Function applied to Normal Element of Unital C*-Algebra is Hermitian iff Function is Real-Valued",
"Definition:Hermitian Element of *-Algebra",
"Norm Preserves Ordering on Positive Elements of C*-Algebra",
"Power of Power of Positive ... |
proofwiki-22197 | Equation of Plane/Normal Form | Let $P$ be a plane.
Let $\LL$ be the normal to $P$ through the origin.
Let:
:the length of $\LL$ be $p$
:the direction cosines of $\LL$ be $l$, $m$ and $n$.
Then $P$ can be defined by the equation:
:$l x + m y + n z = p$ | Let $\mathbf u$ be a unit vector that is perpendicular to $P$ and points towards $P$.
Let $A$ be a point in $P$ such that $A = p \mathbf u$ for some $p \in \mathbb R$.
Therefore, $p$ is the distance from the origin to $P$.
Let $X$ be any other point in $P$.
Let $\mathbf x$ be the vector from the origin to $X$.
Thus, th... | Let $P$ be a [[Definition:Plane|plane]].
Let $\LL$ be the [[Definition:Normal to Surface|normal]] to $P$ through the [[Definition:Origin|origin]].
Let:
:the [[Definition:Length of Line|length]] of $\LL$ be $p$
:the [[Definition:Direction Cosines|direction cosines]] of $\LL$ be $l$, $m$ and $n$.
Then $P$ can be defi... | Let $\mathbf u$ be a [[Definition:Unit Vector|unit vector]] that is [[Definition:Perpendicular|perpendicular]] to $P$ and points towards $P$.
Let $A$ be a [[Definition:Point|point]] in $P$ such that $A = p \mathbf u$ for some $p \in \mathbb R$.
Therefore, $p$ is the [[Definition:Perpendicular Distance between Point a... | Equation of Plane/Normal Form | https://proofwiki.org/wiki/Equation_of_Plane/Normal_Form | https://proofwiki.org/wiki/Equation_of_Plane/Normal_Form | [
"Normal Form of Equation of Plane",
"Equations of Planes"
] | [
"Definition:Plane Surface",
"Definition:Normal to Surface",
"Definition:Coordinate System/Origin",
"Definition:Linear Measure/Length",
"Definition:Direction Cosines",
"Definition:Equation of Geometric Figure"
] | [
"Definition:Unit Vector",
"Definition:Right Angle/Perpendicular",
"Definition:Point",
"Definition:Perpendicular Distance between Point and Plane",
"Definition:Coordinate System/Origin",
"Definition:Point",
"Definition:Position Vector",
"Definition:Coordinate System/Origin",
"Definition:Position Vect... |
proofwiki-22198 | Equation of Plane/Intercept Form | Let $P$ be a plane which intercepts the $x$-axis, $y$-axis and $z$-axis respectively at $\tuple {a, 0, 0}$, $\tuple {0, b, 0}$ and $\tuple {0, 0, c}$, where $a b c \ne 0$.
Then $P$ can be described by the equation:
:$\dfrac x a + \dfrac y b + \dfrac y c = 1$ | We start with the General Equation of Plane:
:$\alpha_1 x + \alpha_2 y + \alpha_3 z = \gamma$
We then plug in the coordinates for each intercept and solve for the corresponding $\alpha$ coefficient.
{{begin-eqn}}
{{eqn | l = \alpha_1 a
| r = \gamma
| c = $\set {a, 0, 0} \in P$
}}
{{eqn | ll=\leadsto
|... | Let $P$ be a [[Definition:Plane|plane]] which [[Definition:Intercept|intercepts]] the [[Definition:X-Axis|$x$-axis]], [[Definition:Y-Axis|$y$-axis]] and [[Definition:Z-Axis|$z$-axis]] respectively at $\tuple {a, 0, 0}$, $\tuple {0, b, 0}$ and $\tuple {0, 0, c}$, where $a b c \ne 0$.
Then $P$ can be described by the e... | We start with the [[General Equation of Plane]]:
:$\alpha_1 x + \alpha_2 y + \alpha_3 z = \gamma$
We then [[Definition:Substitution (Formal Systems)|plug in]] the [[Definition:Coordinates|coordinates]] for each [[Definition:Intercept|intercept]] and [[Definition:Solution of Equation|solve]] for the corresponding $\alp... | Equation of Plane/Intercept Form | https://proofwiki.org/wiki/Equation_of_Plane/Intercept_Form | https://proofwiki.org/wiki/Equation_of_Plane/Intercept_Form | [
"Intercept Form of Equation of Plane",
"Equations of Planes"
] | [
"Definition:Plane Surface",
"Definition:Intercept",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis",
"Definition:Axis/Z-Axis"
] | [
"Equation of Plane/General Equation",
"Definition:Substitution (Formal Systems)",
"Definition:Coordinate System/Coordinate",
"Definition:Intercept",
"Definition:Fiber of Truth/Solution",
"Definition:Coefficient",
"Definition:Substitution (Formal Systems)",
"Equation of Plane/General Equation",
"Defi... |
proofwiki-22199 | Three-Point Form of Equation of Plane/Determinant Form | Let $P_1 := \tuple {x_1, y_1, z_1}$, $P_2 := \tuple {x_2, y_2, z_2}$ and $P_3 := \tuple {x_3, y_3, z_3}$ be non-collinear points in a cartesian $3$-space.
Let $P$ be the plane passing through $P_1$, $P_2$ and $P_3$.
$P$ can be expressed in the form:
:$\begin {vmatrix} x & y & z & 1 \\ x_1 & y_1 & z_1 & 1 \\ x_2 & y_2 &... | === Lemma: $\set {P_1, P_2, P_3} \in P$ ===
Substituting any of $\set {\tuple {x_1, y_1, z_1}, \tuple {x_2, y_2, z_2}, \tuple {x_3, y_3, z_3} }$ for $\tuple {x, y, z}$ satisfies the equation due to Square Matrix with Duplicate Rows has Zero Determinant.
Therefore, the set of points defined by the equation contains $P_1... | Let $P_1 := \tuple {x_1, y_1, z_1}$, $P_2 := \tuple {x_2, y_2, z_2}$ and $P_3 := \tuple {x_3, y_3, z_3}$ be non-[[Definition:Collinear Points|collinear]] [[Definition:Point|points]] in a [[Definition:Cartesian 3-Space|cartesian $3$-space]].
Let $P$ be the [[Definition:Plane|plane]] passing through $P_1$, $P_2$ and $P_... | === Lemma: $\set {P_1, P_2, P_3} \in P$ ===
Substituting any of $\set {\tuple {x_1, y_1, z_1}, \tuple {x_2, y_2, z_2}, \tuple {x_3, y_3, z_3} }$ for $\tuple {x, y, z}$ satisfies the equation due to [[Square Matrix with Duplicate Rows has Zero Determinant]].
Therefore, the [[Definition:Set|set]] of [[Definition:Point|... | Three-Point Form of Equation of Plane/Determinant Form | https://proofwiki.org/wiki/Three-Point_Form_of_Equation_of_Plane/Determinant_Form | https://proofwiki.org/wiki/Three-Point_Form_of_Equation_of_Plane/Determinant_Form | [
"Three-Point Form of Equation of Plane"
] | [
"Definition:Collinear/Points",
"Definition:Point",
"Definition:Cartesian 3-Space",
"Definition:Plane Surface"
] | [
"Square Matrix with Duplicate Rows has Zero Determinant",
"Definition:Set",
"Definition:Point",
"Definition:Set",
"Definition:Point"
] |
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