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 |
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proofwiki-22000 | Spectrum of Element of Unital Commutative Banach Algebra | Let $\struct {A, \norm {\, \cdot \,} }$ be a commutative unital Banach algebra over $\C$.
Let $\Phi_A$ be the spectrum of $A$.
Let $x \in A$.
Let $\map {\sigma_A} x$ be the spectrum of $x$.
Then:
:$\map {\sigma_A} x = \set {\map \phi x : \phi \in \Phi_A}$ | We have $\lambda \in \map {\sigma_A} x$ {{iff}}:
:$\lambda {\mathbf 1}_A - x$ is not invertible.
From Element of Unital Commutative Banach Algebra is Invertible iff not in Kernel of Character, this is the case {{iff}} there exists a character $\phi$ such that:
:$\map \phi {\lambda {\mathbf 1}_A - x} = 0$
From linearity... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Commutative Algebra (Abstract Algebra)|commutative]] [[Definition:Unital Banach Algebra|unital Banach algebra]] over $\C$.
Let $\Phi_A$ be the [[Definition:Spectrum of Banach Algebra|spectrum]] of $A$.
Let $x \in A$.
Let $\map {\sigma_A} x$ be the [[Definiti... | We have $\lambda \in \map {\sigma_A} x$ {{iff}}:
:$\lambda {\mathbf 1}_A - x$ is not [[Definition:Invertible Element|invertible]].
From [[Element of Unital Commutative Banach Algebra is Invertible iff not in Kernel of Character]], this is the case {{iff}} there exists a [[Definition:Character (Banach Algebra)|characte... | Spectrum of Element of Unital Commutative Banach Algebra | https://proofwiki.org/wiki/Spectrum_of_Element_of_Unital_Commutative_Banach_Algebra | https://proofwiki.org/wiki/Spectrum_of_Element_of_Unital_Commutative_Banach_Algebra | [
"Commutative Banach Algebras",
"Unital Banach Algebras",
"Spectrum of Element of Unital Commutative Banach Algebra"
] | [
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Unital Banach Algebra",
"Definition:Spectrum of Banach Algebra",
"Definition:Spectrum (Spectral Theory)/Unital Algebra"
] | [
"Definition:Invertible Element",
"Element of Unital Commutative Banach Algebra is Invertible iff not in Kernel of Character",
"Definition:Character (Banach Algebra)",
"Definition:Linear Transformation",
"Character on Unital Banach Algebra is Unital Algebra Homomorphism",
"Definition:Character (Banach Alge... |
proofwiki-22001 | Spectral Radius of Element of Commutative Banach Algebra | Let $\struct {A, \norm {\, \cdot \,} }$ be a commutative Banach algebra over $\C$.
Let $\Phi_A$ be the spectrum of $A$.
Let $x \in A$.
Let $\map {\sigma_A} x$ be the spectrum of $x$.
Let $\map {r_A} x$ be the spectral radius of $x$.
Then:
:$\ds \map {r_A} x = \sup_{\phi \in \Phi_A} \cmod {\map \phi x}$ | From the definition of the spectral radius, we have:
:$\ds \map {r_A} x = \sup_{\lambda \in \map {\sigma_A} x} \cmod \lambda$
From Spectrum of Element of Unital Commutative Banach Algebra, we have:
:$\map {\sigma_A} x = \set {\map \phi x : \phi \in \Phi_A}$ if $A$ is unital
:$\map {\sigma_A} x = \set {\map \phi x : \p... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Commutative Algebra (Abstract Algebra)|commutative]] [[Definition:Banach Algebra|Banach algebra]] over $\C$.
Let $\Phi_A$ be the [[Definition:Spectrum of Banach Algebra|spectrum]] of $A$.
Let $x \in A$.
Let $\map {\sigma_A} x$ be the [[Definition:Spectrum (S... | From the definition of the [[Definition:Spectral Radius/Banach Algebra|spectral radius]], we have:
:$\ds \map {r_A} x = \sup_{\lambda \in \map {\sigma_A} x} \cmod \lambda$
From [[Spectrum of Element of Unital Commutative Banach Algebra]], we have:
:$\map {\sigma_A} x = \set {\map \phi x : \phi \in \Phi_A}$ if $A$ is ... | Spectral Radius of Element of Commutative Banach Algebra | https://proofwiki.org/wiki/Spectral_Radius_of_Element_of_Commutative_Banach_Algebra | https://proofwiki.org/wiki/Spectral_Radius_of_Element_of_Commutative_Banach_Algebra | [
"Banach Algebras",
"Spectral Radius"
] | [
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Banach Algebra",
"Definition:Spectrum of Banach Algebra",
"Definition:Spectrum (Spectral Theory)/Unital Algebra",
"Definition:Spectral Radius/Banach Algebra"
] | [
"Definition:Spectral Radius/Banach Algebra",
"Spectrum of Element of Unital Commutative Banach Algebra",
"Definition:Unital Banach Algebra",
"Definition:Unital Banach Algebra",
"Category:Banach Algebras",
"Category:Spectral Radius"
] |
proofwiki-22002 | Regression Coefficients of Normally Distributed Random Variable | Let $X$ be a random variable.
Let $x$ be a given value of $X$.
Let $Y$ be a random variable with a normal distribution.
Let the variance $\sigma^2$ of $Y$ (usually unknown) be independent of $x$.
Let $S$ be a sample of $n$ independent pairs of observations $\tuple {x_i, y_i}$ for $i = 1, 2, \ldots, n$.
The maximum like... | The maximum likelihood estimators are obtained by the method of least squares.
That is, the aim is to minimize $\ds \sum_i \paren {y_i - \beta_0 - \beta_1 x_i}^2$.
{{ProofWanted}} | Let $X$ be a [[Definition:Random Variable|random variable]].
Let $x$ be a [[Definition:Given|given value]] of $X$.
Let $Y$ be a [[Definition:Random Variable|random variable]] with a [[Definition:Normal Distribution|normal distribution]].
Let the [[Definition:Variance|variance]] $\sigma^2$ of $Y$ (usually [[Definiti... | The [[Definition:Maximum Likelihood Estimator|maximum likelihood estimators]] are obtained by the [[Definition:Method of Least Squares|method of least squares]].
That is, the aim is to [[Definition:Minimization|minimize]] $\ds \sum_i \paren {y_i - \beta_0 - \beta_1 x_i}^2$.
{{ProofWanted}} | Regression Coefficients of Normally Distributed Random Variable | https://proofwiki.org/wiki/Regression_Coefficients_of_Normally_Distributed_Random_Variable | https://proofwiki.org/wiki/Regression_Coefficients_of_Normally_Distributed_Random_Variable | [
"Regression Coefficients",
"Normal Distribution"
] | [
"Definition:Random Variable",
"Definition:Given",
"Definition:Random Variable",
"Definition:Normal Distribution",
"Definition:Variance",
"Definition:Unknown",
"Definition:Independent Random Variables",
"Definition:Sample",
"Definition:Independent Random Variables",
"Definition:Ordered Pair",
"De... | [
"Definition:Maximum Likelihood Estimator",
"Definition:Method of Least Squares",
"Definition:Minimization"
] |
proofwiki-22003 | Double Root is Root of First Derivative of Equation but not Second Derivative | Let $\EE$ be the equation:
:$\map f x = 0$
where $f: \R \to \R$ is a real function.
Let $\xi$ be a double root of $\EE$.
Then:
:$\xi$ is a root of the equation $\map {f'} x = 0$
:$\xi$ is ''not'' a root of the equation $\map {f' '} x = 0$
where:
:$f': \R \to \R$ is the first derivative of $f$ {{WRT|Differentiation}} $x... | {{DefinitionWanted|Definition about multiplicity of roots on arbitrary functions}}{{ProofWanted}} | Let $\EE$ be the [[Definition:Equation|equation]]:
:$\map f x = 0$
where $f: \R \to \R$ is a [[Definition:Real Function|real function]].
Let $\xi$ be a [[Definition:Double Root|double root]] of $\EE$.
Then:
:$\xi$ is a [[Definition:Root of Equation|root]] of the [[Definition:Equation|equation]] $\map {f'} x = 0$
:$\... | {{DefinitionWanted|Definition about multiplicity of roots on arbitrary functions}}{{ProofWanted}} | Double Root is Root of First Derivative of Equation but not Second Derivative | https://proofwiki.org/wiki/Double_Root_is_Root_of_First_Derivative_of_Equation_but_not_Second_Derivative | https://proofwiki.org/wiki/Double_Root_is_Root_of_First_Derivative_of_Equation_but_not_Second_Derivative | [
"Double Roots",
"Differential Calculus"
] | [
"Definition:Equation",
"Definition:Real Function",
"Definition:Double Root",
"Definition:Root of Equation",
"Definition:Equation",
"Definition:Root of Equation",
"Definition:Equation",
"Definition:Derivative/Real Function/Derivative on Interval",
"Definition:Derivative/Higher Derivatives/Second Deri... | [] |
proofwiki-22004 | Zero Vector in *-Algebra is Hermitian | Let $\struct {A, \ast}$ be a $\ast$-algebra.
Let ${\mathbf 0}_A$ be the zero vector of $A$.
Then we have:
:${\mathbf 0}_A^\ast = {\mathbf 0}_A$ | By $(\text C^\ast 2)$ in the definition of an involution, we have:
:${\mathbf 0}_A^\ast = \paren { {\mathbf 0}_A + {\mathbf 0}_A}^\ast = {\mathbf 0}_A^\ast + {\mathbf 0}_A^\ast$
Hence:
:${\mathbf 0}_A^\ast = {\mathbf 0}_A^\ast - {\mathbf 0}_A^\ast = {\mathbf 0}_A$
{{qed}}
Category:Hermitian Elements of *-Algebras
6hr0r... | Let $\struct {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]].
Let ${\mathbf 0}_A$ be the [[Definition:Zero Vector|zero vector]] of $A$.
Then we have:
:${\mathbf 0}_A^\ast = {\mathbf 0}_A$ | By $(\text C^\ast 2)$ in the definition of an [[Definition:Involution on Algebra|involution]], we have:
:${\mathbf 0}_A^\ast = \paren { {\mathbf 0}_A + {\mathbf 0}_A}^\ast = {\mathbf 0}_A^\ast + {\mathbf 0}_A^\ast$
Hence:
:${\mathbf 0}_A^\ast = {\mathbf 0}_A^\ast - {\mathbf 0}_A^\ast = {\mathbf 0}_A$
{{qed}}
[[Catego... | Zero Vector in *-Algebra is Hermitian | https://proofwiki.org/wiki/Zero_Vector_in_*-Algebra_is_Hermitian | https://proofwiki.org/wiki/Zero_Vector_in_*-Algebra_is_Hermitian | [
"*-Algebras",
"Hermitian Elements of *-Algebras",
"Hermitian Elements of *-Algebras"
] | [
"Definition:*-Algebra",
"Definition:Zero Vector"
] | [
"Definition:Involution on Algebra",
"Category:Hermitian Elements of *-Algebras"
] |
proofwiki-22005 | C* Identity implies Involution is Isometry | Let $\struct {A, \ast}$ be a $\ast$-algebra over $\C$ that is a Banach algebra when given the algebra norm $\norm {\, \cdot \,}$.
Suppose that the $\text C^\ast$ identity:
:$\norm {x x^\ast} = \norm x^2$
holds for all $x \in A$.
Then:
:$\norm x = \norm {x^\ast}$
for each $x \in A$. | Let $x \in A$.
From Zero Vector in *-Algebra is Hermitian, we have:
:${\mathbf 0}_A^\ast = {\mathbf 0}_A$
where ${\mathbf 0}_A$ is the zero vector of $A$.
Hence:
:$\norm x = \norm {x^\ast}$
holds in the case $x = {\mathbf 0}_A$.
Hence take $x \ne {\mathbf 0}_A$ so that $\norm x \ne 0$ by {{NormAxiomVector|1}}.
We ha... | Let $\struct {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]] over $\C$ that is a [[Definition:Banach Algebra|Banach algebra]] when given the [[Definition:Norm on Algebra|algebra norm]] $\norm {\, \cdot \,}$.
Suppose that the [[Definition:C*-Algebra|$\text C^\ast$ identity]]:
:$\norm {x x^\ast} = \norm x^2$
hol... | Let $x \in A$.
From [[Zero Vector in *-Algebra is Hermitian]], we have:
:${\mathbf 0}_A^\ast = {\mathbf 0}_A$
where ${\mathbf 0}_A$ is the [[Definition:Zero Vector|zero vector]] of $A$.
Hence:
:$\norm x = \norm {x^\ast}$
holds in the case $x = {\mathbf 0}_A$.
Hence take $x \ne {\mathbf 0}_A$ so that $\norm x \ne 0... | C* Identity implies Involution is Isometry | https://proofwiki.org/wiki/C*_Identity_implies_Involution_is_Isometry | https://proofwiki.org/wiki/C*_Identity_implies_Involution_is_Isometry | [
"C*-Algebras",
"*-Algebras"
] | [
"Definition:*-Algebra",
"Definition:Banach Algebra",
"Definition:Norm/Algebra",
"Definition:C*-Algebra"
] | [
"Zero Vector in *-Algebra is Hermitian",
"Definition:Zero Vector",
"Definition:Norm/Algebra",
"Definition:Involution on Algebra",
"Category:C*-Algebras",
"Category:*-Algebras"
] |
proofwiki-22006 | Identity Mapping forms Galois Connection | Let $\struct {S, \preceq}$ be an ordered sets.
Let $\operatorname{id}_S : S \to S$ be the identity mapping on $S$.
Then:
:$\tuple {\operatorname{id}_S, \operatorname{id}_S}$ is a Galois connection. | We have
{{begin-eqn}}
{{eqn | q = \forall x, y \in S
| l = x
| o = \preceq
| r = \map {\operatorname{id}_S} y
| c =
}}
{{eqn | ll= \iff
| l = x
| o = \preceq
| r = y
| c = {{Defof|Identity Mapping}}
}}
{{eqn | ll= \iff
| l = \map {\operatorname{id}_S} x
| o ... | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered sets]].
Let $\operatorname{id}_S : S \to S$ be the [[Definition:Identity Mapping|identity mapping]] on $S$.
Then:
:$\tuple {\operatorname{id}_S, \operatorname{id}_S}$ is a [[Definition:Galois Connection|Galois connection]]. | We have
{{begin-eqn}}
{{eqn | q = \forall x, y \in S
| l = x
| o = \preceq
| r = \map {\operatorname{id}_S} y
| c =
}}
{{eqn | ll= \iff
| l = x
| o = \preceq
| r = y
| c = {{Defof|Identity Mapping}}
}}
{{eqn | ll= \iff
| l = \map {\operatorname{id}_S} x
| o ... | Identity Mapping forms Galois Connection | https://proofwiki.org/wiki/Identity_Mapping_forms_Galois_Connection | https://proofwiki.org/wiki/Identity_Mapping_forms_Galois_Connection | [
"Galois Connections"
] | [
"Definition:Ordered Set",
"Definition:Identity Mapping",
"Definition:Galois Connection"
] | [
"Category:Galois Connections"
] |
proofwiki-22007 | Explicit Form for Generated Subalgebra | Let $K$ be a field.
Let $A$ be an algebra over $K$.
Let $S \subseteq A$ be a non-empty set.
Let $K \sqbrk S$ be the subalgebra generated by $A$.
Then:
:$\ds K \sqbrk S = \span \set {x_1^{k_1} x_2^{k_2} \ldots x_n^{k_n} : x_1, \ldots, x_n \in S, \, k_1, \ldots, k_n \ge 1}$ | Let:
:$B = \span \set {x_1^{k_1} x_2^{k_2} \ldots x_n^{k_n} : x_1, \ldots, x_n \in S, \, k_1, \ldots, k_n \ge 1}$
First, for each $x \in S$ we have:
:$x = {\mathbf 1}_K x^1 \in \span \set {x_1^{k_1} x_2^{k_2} \ldots x_n^{k_n} : x_1, \ldots, x_n \in S, \, k_1, \ldots, k_n \ge 1}$
Hence $S \subseteq B$.
We show that $B$ ... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $A$ be an [[Definition:Algebra over Field|algebra]] over $K$.
Let $S \subseteq A$ be a [[Definition:Non-Empty Set|non-empty set]].
Let $K \sqbrk S$ be the [[Definition:Generated Subalgebra|subalgebra generated by $A$]].
Then:
:$\ds K \sqbrk S = \spa... | Let:
:$B = \span \set {x_1^{k_1} x_2^{k_2} \ldots x_n^{k_n} : x_1, \ldots, x_n \in S, \, k_1, \ldots, k_n \ge 1}$
First, for each $x \in S$ we have:
:$x = {\mathbf 1}_K x^1 \in \span \set {x_1^{k_1} x_2^{k_2} \ldots x_n^{k_n} : x_1, \ldots, x_n \in S, \, k_1, \ldots, k_n \ge 1}$
Hence $S \subseteq B$.
We show that $... | Explicit Form for Generated Subalgebra | https://proofwiki.org/wiki/Explicit_Form_for_Generated_Subalgebra | https://proofwiki.org/wiki/Explicit_Form_for_Generated_Subalgebra | [
"Algebras"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Algebra over Field",
"Definition:Non-Empty Set",
"Definition:Generator of Algebra"
] | [
"Definition:Subalgebra",
"Linear Span is Linear Subspace",
"Definition:Linear Subspace",
"Definition:Linear Subspace",
"Definition:Subalgebra",
"Definition:Generator of Algebra",
"Definition:Subalgebra",
"Definition:Linear Subspace",
"Category:Algebras"
] |
proofwiki-22008 | Subalgebra Generated by Commuting Elements is Commutative | Let $K$ be a field.
Let $A$ be an algebra over $K$.
Let $S \subseteq A$ be a non-empty set such that:
:for all $x, y \in S$ we have $x y = y x$.
Let $K \sqbrk S$ be the subalgebra generated by $S$.
Then $K \sqbrk S$ is a commutative algebra. | From Explicit Form for Generated Subalgebra, we have:
:$K \sqbrk S = \span \set {x_1^{k_1} x_2^{k_2} \ldots x_n^{k_n} : x_1, \ldots, x_n \in S, \, k_1, \ldots, k_n \ge 1}$
and $K \sqbrk S$ is a subalgebra of $A$.
Let $x, y \in K \sqbrk S$.
Then:
:$\ds x = \sum_{j \mathop = 1}^N \lambda_j x_{j, 1}^{k_{j, 1} } x_{j, 2}... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $A$ be an [[Definition:Algebra over Field|algebra]] over $K$.
Let $S \subseteq A$ be a [[Definition:Non-Empty Set|non-empty set]] such that:
:for all $x, y \in S$ we have $x y = y x$.
Let $K \sqbrk S$ be the [[Definition:Generated Subalgebra|subalgebr... | From [[Explicit Form for Generated Subalgebra]], we have:
:$K \sqbrk S = \span \set {x_1^{k_1} x_2^{k_2} \ldots x_n^{k_n} : x_1, \ldots, x_n \in S, \, k_1, \ldots, k_n \ge 1}$
and $K \sqbrk S$ is a [[Definition:Subalgebra|subalgebra]] of $A$.
Let $x, y \in K \sqbrk S$.
Then:
:$\ds x = \sum_{j \mathop = 1}^N \lambda... | Subalgebra Generated by Commuting Elements is Commutative | https://proofwiki.org/wiki/Subalgebra_Generated_by_Commuting_Elements_is_Commutative | https://proofwiki.org/wiki/Subalgebra_Generated_by_Commuting_Elements_is_Commutative | [
"Algebras"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Algebra over Field",
"Definition:Non-Empty Set",
"Definition:Generator of Algebra",
"Definition:Commutative Algebra (Abstract Algebra)"
] | [
"Explicit Form for Generated Subalgebra",
"Definition:Subalgebra",
"Definition:Field (Abstract Algebra)",
"Definition:Commutative/Set",
"Definition:Commutative Algebra (Abstract Algebra)",
"Category:Algebras"
] |
proofwiki-22009 | Existence of Maximal Commutative Subalgebra | Let $K$ be a field.
Let $A$ be an algebra over $K$.
Let $S \subseteq A$ be a non-empty set such that:
:for all $x, y \in S$ we have $x y = y x$.
Then there exists a commutative subalgebra of $A$ containing $S$ that is maximal (among all subalgebras) {{WRT}} set inclusion. | Let $\SS$ be the set commutative subalgebras of $A$ containing $S$.
We look to apply Zorn's Lemma to $\struct {\SS, \subseteq}$.
From Subalgebra Generated by Commuting Elements is Commutative, we have $\SS \ne \O$.
Let $\CC$ be a chain in $\struct {\SS, \subseteq}$.
From Union of Chain of Subalgebras is Subalgebra, ... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $A$ be an [[Definition:Algebra over Field|algebra]] over $K$.
Let $S \subseteq A$ be a [[Definition:Non-Empty Set|non-empty set]] such that:
:for all $x, y \in S$ we have $x y = y x$.
Then there exists a [[Definition:Commutative Algebra (Abstract Alg... | Let $\SS$ be the [[Definition:Set|set]] [[Definition:Commutative Algebra (Abstract Algebra)|commutative]] [[Definition:Subalgebra|subalgebras]] of $A$ containing $S$.
We look to apply [[Zorn's Lemma]] to $\struct {\SS, \subseteq}$.
From [[Subalgebra Generated by Commuting Elements is Commutative]], we have $\SS \ne... | Existence of Maximal Commutative Subalgebra | https://proofwiki.org/wiki/Existence_of_Maximal_Commutative_Subalgebra | https://proofwiki.org/wiki/Existence_of_Maximal_Commutative_Subalgebra | [
"Algebras",
"Existence of Maximal Commutative Subalgebra"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Algebra over Field",
"Definition:Non-Empty Set",
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Subalgebra",
"Definition:Maximal/Element",
"Definition:Subalgebra",
"Definition:Subset"
] | [
"Definition:Set",
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Subalgebra",
"Zorn's Lemma",
"Subalgebra Generated by Commuting Elements is Commutative",
"Definition:Chain (Order Theory)",
"Union of Chain of Subalgebras is Subalgebra",
"Definition:Commutative Algebra (Abstract Algeb... |
proofwiki-22010 | Union of Chain of Subalgebras is Subalgebra | Let $K$ be a field.
Let $A$ be an algebra over $K$.
Let $\BB$ be a $\subseteq$-chain of subalgebras of $A$.
Then $\bigcup \BB$ is a subalgebra of $A$.
Further, if each $A \in \BB$ is commutative, then so is $\bigcup \BB$. | From Union of Chain of Submodules is Submodule, $\bigcup \BB$ is a linear subspace of $A$.
Now let $x, y \in \bigcup \BB$.
Then there exists $B_1, B_2 \in \BB$ such that $x \in B_1$ and $y \in B_2$.
Since $\BB$ is a chain, we either have $B_1 \subseteq B_2$ or $B_2 \subseteq B_1$.
Hence we can pick $i \in \set {1, 2}$... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $A$ be an [[Definition:Algebra over Field|algebra]] over $K$.
Let $\BB$ be a [[Definition:Chain (Order Theory)|$\subseteq$-chain]] of [[Definition:Subalgebra|subalgebras]] of $A$.
Then $\bigcup \BB$ is a [[Definition:Subalgebra|subalgebra]] of $A$.
... | From [[Union of Chain of Submodules is Submodule]], $\bigcup \BB$ is a [[Definition:Linear Subspace|linear subspace]] of $A$.
Now let $x, y \in \bigcup \BB$.
Then there exists $B_1, B_2 \in \BB$ such that $x \in B_1$ and $y \in B_2$.
Since $\BB$ is a [[Definition:Chain (Order Theory)|chain]], we either have $B_1 \su... | Union of Chain of Subalgebras is Subalgebra | https://proofwiki.org/wiki/Union_of_Chain_of_Subalgebras_is_Subalgebra | https://proofwiki.org/wiki/Union_of_Chain_of_Subalgebras_is_Subalgebra | [
"Algebras"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Algebra over Field",
"Definition:Chain (Order Theory)",
"Definition:Subalgebra",
"Definition:Subalgebra",
"Definition:Commutative Algebra (Abstract Algebra)"
] | [
"Union of Chain of Submodules is Submodule",
"Definition:Linear Subspace",
"Definition:Chain (Order Theory)",
"Definition:Subalgebra",
"Definition:Subalgebra",
"Definition:Commutative Algebra (Abstract Algebra)",
"Category:Algebras"
] |
proofwiki-22011 | Spectral Radius is Subadditive on Commuting Elements in Banach Algebra | Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$.
Let $x, y \in A$ be commuting elements of $A$.
Let $r_A$ denote spectral radius in $A$.
Then we have:
:$\map {r_A} {x + y} \le \map {r_A} x + \map {r_A} y$ | First take $\struct {A, \norm {\, \cdot \,} }$ to be unital.
From Existence of Maximal Commutative Subalgebra: Unital, there exists a maximal commutative unital subalgebra $B$ containing $x$ and $y$.
From Spectrum of Element in Maximal Commutative Subalgebra of Unital Banach Algebra we have $\map {\sigma_B} x = \map ... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Banach Algebra|Banach algebra]] over $\C$.
Let $x, y \in A$ be [[Definition:Commutative/Set|commuting elements]] of $A$.
Let $r_A$ denote [[Definition:Spectral Radius/Banach Algebra|spectral radius]] in $A$.
Then we have:
:$\map {r_A} {x + y} \le \map {r_A}... | First take $\struct {A, \norm {\, \cdot \,} }$ to be [[Definition:Unital Banach Algebra|unital]].
From [[Existence of Maximal Commutative Subalgebra/Unital|Existence of Maximal Commutative Subalgebra: Unital]], there exists a [[Definition:Maximal Element|maximal]] [[Definition:Commutative Algebra (Abstract Algebra)|c... | Spectral Radius is Subadditive on Commuting Elements in Banach Algebra | https://proofwiki.org/wiki/Spectral_Radius_is_Subadditive_on_Commuting_Elements_in_Banach_Algebra | https://proofwiki.org/wiki/Spectral_Radius_is_Subadditive_on_Commuting_Elements_in_Banach_Algebra | [
"Spectral Radius",
"Banach Algebras"
] | [
"Definition:Banach Algebra",
"Definition:Commutative/Set",
"Definition:Spectral Radius/Banach Algebra"
] | [
"Definition:Unital Banach Algebra",
"Existence of Maximal Commutative Subalgebra/Unital",
"Definition:Maximal/Element",
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Unital Subalgebra",
"Spectrum of Element in Maximal Commutative Subalgebra of Unital Banach Algebra",
"Definition:Spect... |
proofwiki-22012 | Spectral Radius is Multiplicative on Commuting Elements in Banach Algebra | Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$.
Let $x, y \in A$ be commuting elements of $A$.
Let $r_A$ denote spectral radius in $A$.
Then we have:
:$\map {r_A} {x y} \le \map {r_A} x \map {r_A} y$ | First take $\struct {A, \norm {\, \cdot \,} }$ to be unital.
From Existence of Maximal Commutative Subalgebra, there exists a maximal commutative subalgebra $B$ containing $x$ and $y$.
From Maximal Subalgebra in Normed Algebra is Closed, $B$ is closed.
From Spectrum of Element in Maximal Commutative Subalgebra of Uni... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Banach Algebra|Banach algebra]] over $\C$.
Let $x, y \in A$ be [[Definition:Commutative/Set|commuting elements]] of $A$.
Let $r_A$ denote [[Definition:Spectral Radius/Banach Algebra|spectral radius]] in $A$.
Then we have:
:$\map {r_A} {x y} \le \map {r_A} x... | First take $\struct {A, \norm {\, \cdot \,} }$ to be [[Definition:Unital Banach Algebra|unital]].
From [[Existence of Maximal Commutative Subalgebra]], there exists a [[Definition:Maximal Element|maximal]] [[Definition:Commutative Algebra (Abstract Algebra)|commutative]] [[Definition:Subalgebra|subalgebra]] $B$ conta... | Spectral Radius is Multiplicative on Commuting Elements in Banach Algebra | https://proofwiki.org/wiki/Spectral_Radius_is_Multiplicative_on_Commuting_Elements_in_Banach_Algebra | https://proofwiki.org/wiki/Spectral_Radius_is_Multiplicative_on_Commuting_Elements_in_Banach_Algebra | [
"Spectral Radius",
"Banach Algebras"
] | [
"Definition:Banach Algebra",
"Definition:Commutative/Set",
"Definition:Spectral Radius/Banach Algebra"
] | [
"Definition:Unital Banach Algebra",
"Existence of Maximal Commutative Subalgebra",
"Definition:Maximal/Element",
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Subalgebra",
"Maximal Subalgebra in Normed Algebra is Closed",
"Definition:Closed Set",
"Spectrum of Element in Maximal Comm... |
proofwiki-22013 | Character on Unital C*-Algebra is Real at Hermitian Elements | Let $\tuple {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $\phi$ be a character on $A$.
Let $x \in A$ be Hermitian.
Then:
:$\map \phi x \in \R$ | Write:
:$\map \phi x = \alpha + i \beta$
with $\alpha, \beta \in \R$.
Let:
:$x_t = x + i t {\mathbf 1}_A$
for each $t \in \R$.
From Character on Unital Banach Algebra is Unital Algebra Homomorphism, we have $\map \phi { {\mathbf 1}_A} = 1$, and so:
:$\map \phi {x_t} = \map \phi x + i t = \alpha + i \paren {\beta + t... | Let $\tuple {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\phi$ be a [[Definition:Character (Banach Algebra)|character]] on $A$.
Let $x \in A$ be [[Definition:Hermitian Element of *-Algebra|Hermitian]].
Then:
:$\map \phi x \i... | Write:
:$\map \phi x = \alpha + i \beta$
with $\alpha, \beta \in \R$.
Let:
:$x_t = x + i t {\mathbf 1}_A$
for each $t \in \R$.
From [[Character on Unital Banach Algebra is Unital Algebra Homomorphism]], we have $\map \phi { {\mathbf 1}_A} = 1$, and so:
:$\map \phi {x_t} = \map \phi x + i t = \alpha + i \paren {\be... | Character on Unital C*-Algebra is Real at Hermitian Elements | https://proofwiki.org/wiki/Character_on_Unital_C*-Algebra_is_Real_at_Hermitian_Elements | https://proofwiki.org/wiki/Character_on_Unital_C*-Algebra_is_Real_at_Hermitian_Elements | [
"Characters (Banach Algebras)",
"C*-Algebras",
"Hermitian Elements of *-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Character (Banach Algebra)",
"Definition:Hermitian Element of *-Algebra"
] | [
"Character on Unital Banach Algebra is Unital Algebra Homomorphism",
"Character on Banach Algebra is Continuous",
"Definition:Involution on Algebra",
"Definition:Involution on Algebra",
"Bound on Norm of Power of Element in Normed Algebra",
"Category:Characters (Banach Algebras)",
"Category:C*-Algebras"... |
proofwiki-22014 | Character on C*-Algebra is *-Algebra Homomorphism | Let $\tuple {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $\phi$ be a character on $A$.
Then $\phi$ is a $\ast$-algebra homomorphism. | Let $x \in A$.
From Element of *-Algebra Uniquely Decomposes into Hermitian Elements, there exists Hermitian elements $a, b \in A$ such that:
:$x = a + i b$
We have:
{{begin-eqn}}
{{eqn | l = \map \phi {x^\ast}
| r = \map \phi {a^\ast - i b^\ast}
| c = $(\text C^\ast 2)$, $(\text C^\ast 4)$ in the definition of an ... | Let $\tuple {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\phi$ be a [[Definition:Character (Banach Algebra)|character]] on $A$.
Then $\phi$ is a [[Definition:*-Algebra Homomorphism|$\ast$-algebra homomorphism]]. | Let $x \in A$.
From [[Element of *-Algebra Uniquely Decomposes into Hermitian Elements]], there exists [[Definition:Hermitian Element of *-Algebra|Hermitian elements]] $a, b \in A$ such that:
:$x = a + i b$
We have:
{{begin-eqn}}
{{eqn | l = \map \phi {x^\ast}
| r = \map \phi {a^\ast - i b^\ast}
| c = $(\text C^\... | Character on C*-Algebra is *-Algebra Homomorphism | https://proofwiki.org/wiki/Character_on_C*-Algebra_is_*-Algebra_Homomorphism | https://proofwiki.org/wiki/Character_on_C*-Algebra_is_*-Algebra_Homomorphism | [
"Characters (Banach Algebras)",
"C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Character (Banach Algebra)",
"Definition:*-Algebra Homomorphism"
] | [
"Element of *-Algebra Uniquely Decomposes into Hermitian Elements",
"Definition:Hermitian Element of *-Algebra",
"Definition:Involution on Algebra",
"Definition:Hermitian Element of *-Algebra",
"Definition:Linear Functional",
"Character on Unital C*-Algebra is Real at Hermitian Elements",
"Definition:Li... |
proofwiki-22015 | Spectral Radius of Normal Element of C*-Algebra Equal to Norm | 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.
Then $\map {r_A} x = \norm x$. | We first take $x$ Hermitian.
From the $\text C^\ast$ identity, we have:
:$\norm {x^2} = \norm {x x^\ast} = \norm x^2$
for each $x \in A$. | 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]].
Then $\map {r_A} x = \norm x$. | We first take $x$ [[Definition:Hermitian Element of *-Algebra|Hermitian]].
From the [[Definition:C*-Algebra|$\text C^\ast$ identity]], we have:
:$\norm {x^2} = \norm {x x^\ast} = \norm x^2$
for each $x \in A$. | Spectral Radius of Normal Element of C*-Algebra Equal to Norm | https://proofwiki.org/wiki/Spectral_Radius_of_Normal_Element_of_C*-Algebra_Equal_to_Norm | https://proofwiki.org/wiki/Spectral_Radius_of_Normal_Element_of_C*-Algebra_Equal_to_Norm | [
"C*-Algebras",
"Normal Elements of *-Algebras",
"Spectral Radius",
"Spectral Radius of Normal Element of C*-Algebra Equal to Norm"
] | [
"Definition:C*-Algebra",
"Definition:Spectral Radius/Banach Algebra",
"Definition:Normal Element of *-Algebra"
] | [
"Definition:Hermitian Element of *-Algebra",
"Definition:C*-Algebra",
"Definition:Hermitian Element of *-Algebra",
"Definition:Hermitian Element of *-Algebra",
"Definition:C*-Algebra"
] |
proofwiki-22016 | Norm on C*-Algebra is Unique | Let $\struct {A, \ast}$ be a $\ast$-algebra over $\C$.
Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ be algebra norms on $\struct {A, \ast}$ such that:
:$\struct {A, \ast, \norm {\, \cdot \,}_1}$ and $\struct {A, \ast, \norm {\, \cdot \,}_2}$ are $\text C^\ast$-algebras.
Then:
:$\norm x_1 = \norm x_2$
for ea... | Note that the spectrum of $x \in A$ is defined independently of the algebra norm on $A$, depending only on the invertible elements of $A$.
With that:
{{begin-eqn}}
{{eqn | l = \norm x_1^2
| r = \norm {x x^\ast}_1
| c = {{Defof|C*-Algebra}}
}}
{{eqn | r = \map {r_A} {x x^\ast}
| c = Product of Element in *-Star Al... | Let $\struct {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]] over $\C$.
Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ be [[Definition:Norm on Algebra|algebra norms]] on $\struct {A, \ast}$ such that:
:$\struct {A, \ast, \norm {\, \cdot \,}_1}$ and $\struct {A, \ast, \norm {\, \cdot \,}_2}$ are [[Def... | Note that the [[Definition:Spectrum (Spectral Theory)|spectrum]] of $x \in A$ is defined independently of the [[Definition:Norm on Algebra|algebra norm]] on $A$, depending only on the [[Definition:Invertible Element|invertible elements]] of $A$.
With that:
{{begin-eqn}}
{{eqn | l = \norm x_1^2
| r = \norm {x x^\ast}... | Norm on C*-Algebra is Unique | https://proofwiki.org/wiki/Norm_on_C*-Algebra_is_Unique | https://proofwiki.org/wiki/Norm_on_C*-Algebra_is_Unique | [
"C*-Algebras"
] | [
"Definition:*-Algebra",
"Definition:Norm/Algebra",
"Definition:C*-Algebra"
] | [
"Definition:Spectrum (Spectral Theory)",
"Definition:Norm/Algebra",
"Definition:Invertible Element",
"Product of Element in *-Star Algebra with its Star is Hermitian",
"Spectral Radius of Normal Element of C*-Algebra Equal to Norm",
"Product of Element in *-Star Algebra with its Star is Hermitian",
"Spe... |
proofwiki-22017 | Norm of Element of C*-Algebra as Supremum over Closed Unit Ball | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $c \in A$.
Then:
:$\ds \norm c = \sup_{\norm b \le 1} \norm {c b} = \sup_{\norm b \le 1} \norm {b c}$ | Note that if $c = {\mathbf 0}_A$, we have $\norm c = 0$ and:
:$\norm {c b} = \norm {b c} = \norm { {\mathbf 0}_A} = 0$
for each $b \in A$.
Hence we have the theorem in the case $c = {\mathbf 0}_A$.
Now take $c \ne {\mathbf 0}_A$.
From the definition of an algebra norm, we have:
:$\norm {c b} \le \norm c \norm b$
and:
:... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $c \in A$.
Then:
:$\ds \norm c = \sup_{\norm b \le 1} \norm {c b} = \sup_{\norm b \le 1} \norm {b c}$ | Note that if $c = {\mathbf 0}_A$, we have $\norm c = 0$ and:
:$\norm {c b} = \norm {b c} = \norm { {\mathbf 0}_A} = 0$
for each $b \in A$.
Hence we have the theorem in the case $c = {\mathbf 0}_A$.
Now take $c \ne {\mathbf 0}_A$.
From the definition of an [[Definition:Norm on Algebra|algebra norm]], we have:
:$\norm... | Norm of Element of C*-Algebra as Supremum over Closed Unit Ball | https://proofwiki.org/wiki/Norm_of_Element_of_C*-Algebra_as_Supremum_over_Closed_Unit_Ball | https://proofwiki.org/wiki/Norm_of_Element_of_C*-Algebra_as_Supremum_over_Closed_Unit_Ball | [
"C*-Algebras"
] | [
"Definition:C*-Algebra"
] | [
"Definition:Norm/Algebra",
"Definition:Banach *-Algebra",
"Definition:C*-Algebra",
"Definition:C*-Algebra"
] |
proofwiki-22018 | Norms of Double Centralizer of C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $\tuple {L, R}$ be a double centralizer of $A$.
Then $\norm L_{\map B A} = \norm R_{\map B A}$, where $\norm L_{\map B A}$ and $\norm R_{\map B A}$ denotes the norm of a bounded linear transformation. | For each $a, b \in A$, we have:
{{begin-eqn}}
{{eqn | l = \norm {a \map L b}
| r = \norm {\map R a b}
| c = {{Defof|Double Centralizer of C*-Algebra}}
}}
{{eqn | o = \le
| r = \norm {\map R a} \norm b
| c = {{Defof|Norm on Algebra}}
}}
{{eqn | o = \le
| r = \norm R_{\map B A} \norm a \norm b
| c = Fundamen... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\tuple {L, R}$ be a [[Definition:Double Centralizer of C*-Algebra|double centralizer]] of $A$.
Then $\norm L_{\map B A} = \norm R_{\map B A}$, where $\norm L_{\map B A}$ and $\norm R_{\map B A}$ denotes the [[De... | For each $a, b \in A$, we have:
{{begin-eqn}}
{{eqn | l = \norm {a \map L b}
| r = \norm {\map R a b}
| c = {{Defof|Double Centralizer of C*-Algebra}}
}}
{{eqn | o = \le
| r = \norm {\map R a} \norm b
| c = {{Defof|Norm on Algebra}}
}}
{{eqn | o = \le
| r = \norm R_{\map B A} \norm a \norm b
| c = [[Fundam... | Norms of Double Centralizer of C*-Algebra | https://proofwiki.org/wiki/Norms_of_Double_Centralizer_of_C*-Algebra | https://proofwiki.org/wiki/Norms_of_Double_Centralizer_of_C*-Algebra | [
"Multiplier Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Double Centralizer of C*-Algebra",
"Definition:Norm/Bounded Linear Transformation"
] | [
"Fundamental Property of Norm on Bounded Linear Transformation",
"Norm of Element of C*-Algebra as Supremum over Closed Unit Ball",
"Definition:Norm/Bounded Linear Transformation",
"Fundamental Property of Norm on Bounded Linear Transformation",
"Norm of Element of C*-Algebra as Supremum over Closed Unit Ba... |
proofwiki-22019 | Double Centralizer Generated by Element of C*-Algebra is Double Centralizer | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $c \in A$.
Define $L_c : A \to A$ and $R_c : A \to A$ by:
:$\map {L_c} a = c a$
and:
:$\map {R_c} a = a c$
for each $a \in A$.
Then $\tuple {L_c, R_c}$ is a double centralizer of $A$. | First, for each $a \in A$ we have:
:$\norm {\map {L_c} a} \le \norm c \norm a$
and:
:$\norm {\map {R_c} a} \le \norm a \norm c$
by the definition of an algebra norm.
Now, for $a, b \in A$ we have:
{{begin-eqn}}
{{eqn | l = \map {L_c} {a b}
| r = c a b
}}
{{eqn | r = \paren {c a} b
}}
{{eqn | r = \map {L_c} a b
}}
{{e... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $c \in A$.
Define $L_c : A \to A$ and $R_c : A \to A$ by:
:$\map {L_c} a = c a$
and:
:$\map {R_c} a = a c$
for each $a \in A$.
Then $\tuple {L_c, R_c}$ is a [[Definition:Double Centralizer of C*-Algebra|double ... | First, for each $a \in A$ we have:
:$\norm {\map {L_c} a} \le \norm c \norm a$
and:
:$\norm {\map {R_c} a} \le \norm a \norm c$
by the definition of an [[Definition:Norm on Algebra|algebra norm]].
Now, for $a, b \in A$ we have:
{{begin-eqn}}
{{eqn | l = \map {L_c} {a b}
| r = c a b
}}
{{eqn | r = \paren {c a} b
}}
{... | Double Centralizer Generated by Element of C*-Algebra is Double Centralizer | https://proofwiki.org/wiki/Double_Centralizer_Generated_by_Element_of_C*-Algebra_is_Double_Centralizer | https://proofwiki.org/wiki/Double_Centralizer_Generated_by_Element_of_C*-Algebra_is_Double_Centralizer | [
"Multiplier Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Double Centralizer of C*-Algebra"
] | [
"Definition:Norm/Algebra",
"Definition:Double Centralizer of C*-Algebra"
] |
proofwiki-22020 | Norm of Double Centralizer Generated by Element of C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $c \in A$.
Let $\tuple {L_c, R_c}$ be the double centralizer generated by $c$.
Then:
:$\norm {L_c}_{\map B A} = \norm {R_c}_{\map B A} = \norm c$
where $\norm {L_c}_{\map B A}$ and $\norm {R_c}_{\map B A}$ denotes the norm of a bounded linea... | From Norm of Element of C*-Algebra as Supremum over Closed Unit Ball we have:
:$\ds \norm c = \sup_{\norm b \le 1} \norm {c b} = \sup_{\norm b \le 1} \norm {b c}$
From the definition of the double centralizer generated by $c$, we have:
:$\ds \norm c = \sup_{\norm b \le 1} \norm {\map {L_c} b} = \sup_{\norm b \le 1} \no... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $c \in A$.
Let $\tuple {L_c, R_c}$ be the [[Definition:Double Centralizer of C*-Algebra|double centralizer generated by $c$]].
Then:
:$\norm {L_c}_{\map B A} = \norm {R_c}_{\map B A} = \norm c$
where $\norm {L_c... | From [[Norm of Element of C*-Algebra as Supremum over Closed Unit Ball]] we have:
:$\ds \norm c = \sup_{\norm b \le 1} \norm {c b} = \sup_{\norm b \le 1} \norm {b c}$
From the definition of the [[Definition:Double Centralizer of C*-Algebra|double centralizer generated by $c$]], we have:
:$\ds \norm c = \sup_{\norm b \... | Norm of Double Centralizer Generated by Element of C*-Algebra | https://proofwiki.org/wiki/Norm_of_Double_Centralizer_Generated_by_Element_of_C*-Algebra | https://proofwiki.org/wiki/Norm_of_Double_Centralizer_Generated_by_Element_of_C*-Algebra | [
"Multiplier Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Double Centralizer of C*-Algebra",
"Definition:Norm/Bounded Linear Transformation"
] | [
"Norm of Element of C*-Algebra as Supremum over Closed Unit Ball",
"Definition:Double Centralizer of C*-Algebra",
"Definition:Norm/Bounded Linear Transformation"
] |
proofwiki-22021 | Sufficient Condition for C* Identity | Let $\tuple {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$.
Let $\ast$ be an involution on $A$ such that:
:$\norm a^2 \le \norm {a a^\ast}$
Then $\tuple {A, \ast, \norm {\, \cdot \,} }$ is a $\text C^\ast$-algebra. | Let $a \in A$.
From Zero Vector in *-Algebra is Hermitian and {{NormAxiomVector|1}}, we have:
:$\norm a^2 = \norm {a a^\ast}$
for $a = {\mathbf 0}_A$.
Now take $a \ne {\mathbf 0}_A$ so that $\norm a \ne 0$.
From the definition of an algebra norm:
:$\norm {a a^\ast} \le \norm a \norm {a^\ast}$
Hence:
:$\norm a^2 \le \n... | Let $\tuple {A, \norm {\, \cdot \,} }$ be a [[Definition:Banach Algebra|Banach algebra]] over $\C$.
Let $\ast$ be an [[Definition:Involution on Algebra|involution]] on $A$ such that:
:$\norm a^2 \le \norm {a a^\ast}$
Then $\tuple {A, \ast, \norm {\, \cdot \,} }$ is a [[Definition:C*-Algebra|$\text C^\ast$-algebra]]. | Let $a \in A$.
From [[Zero Vector in *-Algebra is Hermitian]] and {{NormAxiomVector|1}}, we have:
:$\norm a^2 = \norm {a a^\ast}$
for $a = {\mathbf 0}_A$.
Now take $a \ne {\mathbf 0}_A$ so that $\norm a \ne 0$.
From the definition of an [[Definition:Norm on Algebra|algebra norm]]:
:$\norm {a a^\ast} \le \norm a \no... | Sufficient Condition for C* Identity | https://proofwiki.org/wiki/Sufficient_Condition_for_C*_Identity | https://proofwiki.org/wiki/Sufficient_Condition_for_C*_Identity | [
"C*-Algebras"
] | [
"Definition:Banach Algebra",
"Definition:Involution on Algebra",
"Definition:C*-Algebra"
] | [
"Zero Vector in *-Algebra is Hermitian",
"Definition:Norm/Algebra",
"Definition:C*-Algebra",
"Definition:C*-Algebra",
"Category:C*-Algebras"
] |
proofwiki-22022 | Equivalent Norms on Direct Product of Normed Vector Spaces | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces over $\GF$.
Let $\struct {X \times Y, \norm {\, \cdot \,}_{X \times Y} }$ be the direct product of $X$ and $Y$ with the direct product norm.
Let $p \ge 1$.
Let $\norm {\, \cdot \,}_p... | We first show that $\norm {\, \cdot \,}'$ is a norm.
Towards proving {{NormAxiomVector|1}}, suppose that $\norm {\tuple {x, y} }' = 0$.
Then we have $\norm {\tuple {\norm x_X, \norm y_Y} }_p = 0$.
Hence $\norm x_X = 0$ and $\norm y_Y = 0$ by {{NormAxiomVector|1}}.
From {{NormAxiomVector|1}}, we have $x = {\mathbf 0}_X... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be [[Definition:Normed Vector Space|normed vector spaces]] over $\GF$.
Let $\struct {X \times Y, \norm {\, \cdot \,}_{X \times Y} }$ be the [[Definition:Direct Product of Vector Spaces|direct product]] of $... | We first show that $\norm {\, \cdot \,}'$ is a [[Definition:Norm on Vector Space|norm]].
Towards proving {{NormAxiomVector|1}}, suppose that $\norm {\tuple {x, y} }' = 0$.
Then we have $\norm {\tuple {\norm x_X, \norm y_Y} }_p = 0$.
Hence $\norm x_X = 0$ and $\norm y_Y = 0$ by {{NormAxiomVector|1}}.
From {{NormAxi... | Equivalent Norms on Direct Product of Normed Vector Spaces | https://proofwiki.org/wiki/Equivalent_Norms_on_Direct_Product_of_Normed_Vector_Spaces | https://proofwiki.org/wiki/Equivalent_Norms_on_Direct_Product_of_Normed_Vector_Spaces | [
"Direct Product of Vector Spaces",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Direct Product of Vector Spaces",
"Definition:Direct Product Norm",
"Definition:p-Norm",
"Definition:Norm/Vector Space",
"Definition:Equivalence of Norms"
] | [
"Definition:Norm/Vector Space",
"Definition:Norm/Vector Space",
"Norms on Finite-Dimensional Real Vector Space are Equivalent",
"Definition:Equivalence of Norms",
"Category:Direct Product of Vector Spaces",
"Category:Normed Vector Spaces"
] |
proofwiki-22023 | Multiplication on Normed Algebra is Continuous | Let $\GF \in \set {\R, \C}$.
Let $\struct {A, \norm {\, \cdot \,} }$ be a normed algebra over $\GF$.
Let $\struct {A^2, \norm {\, \cdot \,}_{A^2} }$ be the direct product of $A$ with itself equipped with the direct product norm.
Define $m : A^2 \to A$ by:
:$\map m {a, b} = a b$
for each $\tuple {a, b} \in A^2$.
Then ... | Fix $\tuple {a, b} \in A^2$.
Let $\tuple {a', b'} \in A^2$.
Then we have:
{{begin-eqn}}
{{eqn | l = \norm {\map m {a, b} - \map m {a', b'} }
| r = \norm {a b - a' b'}
}}
{{eqn | r = \norm {a b - a' b + a' b - a' b'}
}}
{{eqn | r = \norm {b \paren {a - a'} + a' \paren {b - b'} }
}}
{{eqn | o = \le
| r = \norm {b \... | Let $\GF \in \set {\R, \C}$.
Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Normed Algebra|normed algebra]] over $\GF$.
Let $\struct {A^2, \norm {\, \cdot \,}_{A^2} }$ be the [[Definition:Direct Product of Vector Spaces|direct product]] of $A$ with itself equipped with the [[Definition:Direct Product Norm... | Fix $\tuple {a, b} \in A^2$.
Let $\tuple {a', b'} \in A^2$.
Then we have:
{{begin-eqn}}
{{eqn | l = \norm {\map m {a, b} - \map m {a', b'} }
| r = \norm {a b - a' b'}
}}
{{eqn | r = \norm {a b - a' b + a' b - a' b'}
}}
{{eqn | r = \norm {b \paren {a - a'} + a' \paren {b - b'} }
}}
{{eqn | o = \le
| r = \norm {b... | Multiplication on Normed Algebra is Continuous | https://proofwiki.org/wiki/Multiplication_on_Normed_Algebra_is_Continuous | https://proofwiki.org/wiki/Multiplication_on_Normed_Algebra_is_Continuous | [
"Normed Algebras"
] | [
"Definition:Normed Algebra",
"Definition:Direct Product of Vector Spaces",
"Definition:Direct Product Norm",
"Definition:Continuous Mapping"
] | [
"Reverse Triangle Inequality/Normed Vector Space",
"Definition:Continuous Mapping (Normed Vector Space)/Point",
"Reverse Triangle Inequality/Normed Vector Space",
"Definition:Continuous Mapping (Normed Vector Space)/Point",
"Definition:Continuous Mapping (Normed Vector Space)/Point",
"Definition:Continuou... |
proofwiki-22024 | Multiplier Algebra is Unital C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $\map M A$ be the set of double centralizers of $A$.
Define:
:$\tuple {L_1, R_1} +_{\map M A} \tuple {L_2, R_2} = \tuple {L_1 +_{\map B A} L_2, R_1 +_{\map B A} R_2}$
:$\lambda \circ_{\map M A} \tuple {L_1, R_1} = \tuple {\lambda \circ_{\m... | === Proof that $\map M A$ is a vector space ===
We show that $\map M A$ is a subspace of the direct product $\map B A \times \map B A$.
We use the One-Step Vector Subspace Test.
From Double Centralizer Generated by Element of C*-Algebra is Double Centralizer, we have $\map M A \ne \O$.
Let $\tuple {L_1, R_1}, \tuple {... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\map M A$ be the [[Definition:Set|set]] of [[Definition:Double Centralizer of C*-Algebra|double centralizers]] of $A$.
Define:
:$\tuple {L_1, R_1} +_{\map M A} \tuple {L_2, R_2} = \tuple {L_1 +_{\map B A} L_2, ... | === Proof that $\map M A$ is a [[Definition:Vector Space|vector space]] ===
We show that $\map M A$ is a [[Definition:Vector Subspace|subspace]] of the [[Definition:Direct Product of Vector Spaces|direct product]] $\map B A \times \map B A$.
We use the [[One-Step Vector Subspace Test]].
From [[Double Centralizer Ge... | Multiplier Algebra is Unital C*-Algebra | https://proofwiki.org/wiki/Multiplier_Algebra_is_Unital_C*-Algebra | https://proofwiki.org/wiki/Multiplier_Algebra_is_Unital_C*-Algebra | [
"Multiplier Algebras",
"C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Set",
"Definition:Double Centralizer of C*-Algebra",
"Definition:Space of Bounded Linear Transformations",
"Definition:Norm/Bounded Linear Transformation",
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra"
] | [
"Definition:Vector Space",
"Definition:Vector Subspace",
"Definition:Direct Product of Vector Spaces",
"One-Step Vector Subspace Test",
"Double Centralizer Generated by Element of C*-Algebra is Double Centralizer",
"Definition:Double Centralizer of C*-Algebra",
"Definition:Associative Operation",
"Def... |
proofwiki-22025 | Galois Connection is Unique for Given Lower Adjoint | Let $\struct {S, \preceq}$ and $\struct {T, \precsim}$ be ordered sets.
Let $g: S \to T$, $d: T \to S$ be mappings such that $\tuple{g, d}$ is a Galois connection.
Then:
:$\tuple{g, d}$ is the unique Galois connection such that $d$ is the lower adjoint | Let $g^\prime: S \to T$ be a mapping such that $\tuple{g^\prime, d}$ is a Galois connection.
We have:
{{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {g^\prime} x
| r = \map \max {d^{-1} \sqbrk {x^\preceq} }
| c = Galois Connection is Expressed by Maximum
}}
{{eqn | r = \map g x
| c = Galo... | Let $\struct {S, \preceq}$ and $\struct {T, \precsim}$ be [[Definition:Ordered Set|ordered sets]].
Let $g: S \to T$, $d: T \to S$ be [[Definition:Mapping|mappings]] such that $\tuple{g, d}$ is a [[Definition:Galois Connection|Galois connection]].
Then:
:$\tuple{g, d}$ is the [[Definition:Unique|unique]] [[Definition... | Let $g^\prime: S \to T$ be a [[Definition:Mapping|mapping]] such that $\tuple{g^\prime, d}$ is a [[Definition:Galois Connection|Galois connection]].
We have:
{{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {g^\prime} x
| r = \map \max {d^{-1} \sqbrk {x^\preceq} }
| c = [[Galois Connection is E... | Galois Connection is Unique for Given Lower Adjoint | https://proofwiki.org/wiki/Galois_Connection_is_Unique_for_Given_Lower_Adjoint | https://proofwiki.org/wiki/Galois_Connection_is_Unique_for_Given_Lower_Adjoint | [
"Galois Connections"
] | [
"Definition:Ordered Set",
"Definition:Mapping",
"Definition:Galois Connection",
"Definition:Unique",
"Definition:Galois Connection",
"Definition:Galois Connection/Lower Adjoint"
] | [
"Definition:Mapping",
"Definition:Galois Connection",
"Galois Connection is Expressed by Maximum",
"Galois Connection is Expressed by Maximum",
"Definition:Greatest Element/Subset",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Lower Closure/Element",
"Equality of Mappings",
"Category:... |
proofwiki-22026 | Galois Connection is Unique for Given Upper Adjoint | Let $\struct {S, \preceq}$ and $\struct {T, \precsim}$ be ordered sets.
Let $g: S \to T$, $d: T \to S$ be mappings such that $\tuple{g, d}$ is a Galois connection.
Then:
:$\tuple{g, d}$ is the unique Galois connection such that $g$ is the upper adjoint | Let $d^\prime: T \to S$ be a mapping such that $\tuple{g, d^\prime}$ is a Galois connection.
We have:
{{begin-eqn}}
{{eqn | q = \forall y \in T
| l = \map {d^\prime} y
| r = \map \min {g^{-1} \sqbrk {t^\succsim} }
| c = Galois Connection is Expressed by Minimum
}}
{{eqn | r = \map d x
| c = Gal... | Let $\struct {S, \preceq}$ and $\struct {T, \precsim}$ be [[Definition:Ordered Set|ordered sets]].
Let $g: S \to T$, $d: T \to S$ be [[Definition:Mapping|mappings]] such that $\tuple{g, d}$ is a [[Definition:Galois Connection|Galois connection]].
Then:
:$\tuple{g, d}$ is the [[Definition:Unique|unique]] [[Definition... | Let $d^\prime: T \to S$ be a [[Definition:Mapping|mapping]] such that $\tuple{g, d^\prime}$ is a [[Definition:Galois Connection|Galois connection]].
We have:
{{begin-eqn}}
{{eqn | q = \forall y \in T
| l = \map {d^\prime} y
| r = \map \min {g^{-1} \sqbrk {t^\succsim} }
| c = [[Galois Connection is ... | Galois Connection is Unique for Given Upper Adjoint | https://proofwiki.org/wiki/Galois_Connection_is_Unique_for_Given_Upper_Adjoint | https://proofwiki.org/wiki/Galois_Connection_is_Unique_for_Given_Upper_Adjoint | [
"Galois Connections"
] | [
"Definition:Ordered Set",
"Definition:Mapping",
"Definition:Galois Connection",
"Definition:Unique",
"Definition:Galois Connection",
"Definition:Galois Connection/Upper Adjoint"
] | [
"Definition:Mapping",
"Definition:Galois Connection",
"Galois Connection is Expressed by Minimum",
"Galois Connection is Expressed by Minimum",
"Definition:Smallest Element/Subset",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Upper Closure/Element",
"Equality of Mappings",
"Category:... |
proofwiki-22027 | Extension of Continuous Complex-Valued Function Vanishing at Infinity to Alexandroff Extension is Continuous | Let $X$ be a locally compact Hausdorff space.
Let $X^\ast = X \cup \set p$ be the Alexandroff extension of $X$.
Let $f : X \to \C$ be a continuous complex-valued function vanishing at infinity.
Define $f^\ast : X^\ast \to \C$ by taking:
:$\map {f^\ast} x = \begin{cases}\map f x & x \in X \\ 0 & x = p\end{cases}$
for ea... | Since $f$ is continuous at each $x \in X$, we have:
:for each $x \in X$:
::for every open neighborhood $U_2$ of $\map f x \in \C$, there exists an open neighborhood $U_1$ of $x \in X$ such that $f \sqbrk {U_1} \subseteq U_2$.
From the definition of the topology on $X^\ast$:
:every open set in $X$ is open in $X^\ast$.
H... | Let $X$ be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]].
Let $X^\ast = X \cup \set p$ be the [[Definition:Alexandroff Extension|Alexandroff extension]] of $X$.
Let $f : X \to \C$ be a [[Definition:Continuous Mapping|continuous]] [[Definition:Complex-Valued Function Vanishing at Inf... | Since $f$ is [[Definition:Continuous at Point of Topological Space|continuous]] at each $x \in X$, we have:
:for each $x \in X$:
::for every [[Definition:Open Neighborhood|open neighborhood]] $U_2$ of $\map f x \in \C$, there exists an [[Definition:Open Neighborhood|open neighborhood]] $U_1$ of $x \in X$ such that $f \... | Extension of Continuous Complex-Valued Function Vanishing at Infinity to Alexandroff Extension is Continuous | https://proofwiki.org/wiki/Extension_of_Continuous_Complex-Valued_Function_Vanishing_at_Infinity_to_Alexandroff_Extension_is_Continuous | https://proofwiki.org/wiki/Extension_of_Continuous_Complex-Valued_Function_Vanishing_at_Infinity_to_Alexandroff_Extension_is_Continuous | [
"Extension of Continuous Complex-Valued Function Vanishing at Infinity to Alexandroff Extension is Continuous",
"Alexandroff Extensions",
"Complex-Valued Functions Vanishing at Infinity"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:Alexandroff Extension",
"Definition:Continuous Mapping",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Continuous Function"
] | [
"Definition:Continuous Mapping (Topology)/Point",
"Definition:Open Neighborhood",
"Definition:Open Neighborhood",
"Definition:Topology",
"Definition:Open Set",
"Definition:Open Set",
"Definition:Open Neighborhood",
"Definition:Continuous Mapping (Topology)/Point",
"Definition:Continuous Mapping (Top... |
proofwiki-22028 | Bounded Continuous Functions on Topological Space form Banach Space | Let $\GF \in \set {\R, \C}$.
Let $X$ be a topological space.
Let $Y$ be a Banach space over $\GF$.
Let $\map {\CC_b} {X; Y}$ be the space of bounded continuous functions on $X$ valued in $Y$.
Let $\norm {\,\cdot\,}_\infty$ be the supremum norm on $\CC$.
Then $\struct {\map {\CC_b} {X; Y}, \norm {\,\cdot\,}_\infty}$ i... | We first show that $\map {\CC_b} {X; Y}$ is a vector space over $\GF$.
We have that the set of continuous mappings $X \to Y$ is a subset of the set $Y^X$ of ''all'' mappings $X \to Y$.
Therefore by Vector Space of All Mappings is Vector Space, we need only show that $\map {\CC_b} {X; Y}$ is a subspace of $Y^X$.
First, ... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Topological Space|topological space]].
Let $Y$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $\map {\CC_b} {X; Y}$ be the [[Definition:Space of Bounded Continuous Functions on Topological Space|space of bounded continuous functions on $X$ valued... | We first show that $\map {\CC_b} {X; Y}$ is a [[Definition:Vector Space|vector space]] over $\GF$.
We have that the [[Definition:Set|set]] of [[Definition:Everywhere Continuous Mapping (Topology)|continuous mappings]] $X \to Y$ is a [[Definition:Subset|subset]] of the [[Definition:Set|set]] $Y^X$ of ''all'' [[Definiti... | Bounded Continuous Functions on Topological Space form Banach Space | https://proofwiki.org/wiki/Bounded_Continuous_Functions_on_Topological_Space_form_Banach_Space | https://proofwiki.org/wiki/Bounded_Continuous_Functions_on_Topological_Space_form_Banach_Space | [
"Functional Analysis"
] | [
"Definition:Topological Space",
"Definition:Banach Space",
"Definition:Space of Bounded Continuous Functions on Topological Space",
"Definition:Supremum Norm",
"Definition:Banach Space"
] | [
"Definition:Vector Space",
"Definition:Set",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Subset",
"Definition:Set",
"Definition:Mapping",
"Vector Space of All Mappings is Vector Space",
"Definition:Vector Subspace",
"Constant Function is Continuous",
"Definition:Continuous M... |
proofwiki-22029 | Complex Numbers form Unital C*-Algebra | Let $\C$ be the complex numbers realised as a normed algebra (this is possible from Complex Numbers form Algebra).
Let $\overline {\, \cdot \,}$ be complex conjugation.
Then $\tuple {\C, \overline {\, \cdot \,} }$ is a unital $\text C^\ast$-algebra. | From Complex Plane is Banach Space, $\C$ is a Banach space.
Further, we have:
:$\cmod 1 = 1$
Since $\C$ is also a normed algebra, it therefore a unital Banach algebra.
We prove that complex conjugation satisfies the properties of an involution as well as the $\text C^\ast$ identity. | Let $\C$ be the [[Definition:Complex Number|complex numbers]] realised as a [[Definition:Normed Algebra|normed algebra]] (this is possible from [[Complex Numbers form Algebra]]).
Let $\overline {\, \cdot \,}$ be [[Definition:Complex Conjugation|complex conjugation]].
Then $\tuple {\C, \overline {\, \cdot \,} }$ is a... | From [[Complex Plane is Banach Space]], $\C$ is a [[Definition:Banach Space|Banach space]].
Further, we have:
:$\cmod 1 = 1$
Since $\C$ is also a [[Definition:Normed Algebra|normed algebra]], it therefore a [[Definition:Unital Banach Algebra|unital Banach algebra]].
We prove that [[Definition:Complex Conjugation|com... | Complex Numbers form Unital C*-Algebra | https://proofwiki.org/wiki/Complex_Numbers_form_Unital_C*-Algebra | https://proofwiki.org/wiki/Complex_Numbers_form_Unital_C*-Algebra | [
"C*-Algebras",
"Complex Numbers"
] | [
"Definition:Complex Number",
"Definition:Normed Algebra",
"Complex Numbers form Algebra",
"Definition:Complex Conjugate/Complex Conjugation",
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra"
] | [
"Complex Plane is Banach Space",
"Definition:Banach Space",
"Definition:Normed Algebra",
"Definition:Unital Banach Algebra",
"Definition:Complex Conjugate/Complex Conjugation",
"Definition:Involution on Algebra",
"Definition:C* Identity",
"Definition:Unital Banach Algebra",
"Definition:C* Identity",... |
proofwiki-22030 | Bounded Complex-Valued Continuous Functions on Topological Space form Unital C*-Algebra | Let $X$ be a topological space.
Let $\map {\CC_b} {X; \C} := \map {\CC_b} X$ be the space of bounded continuous functions on $X$ valued in $\C$.
Let $\norm {\,\cdot\,}_\infty$ be the supremum norm on $\map {\CC_b} X$.
For each $f \in \map {\CC_b} X$, define $\overline f : X \to \C$ by:
:$\map {\overline f} x = \overli... | From Bounded Continuous Functions on Topological Space form Banach Space, $\struct {\map {\CC_b} X, \norm {\,\cdot\,}_\infty}$ is a Banach space over $\C$.
From Product of Continuous Functions on Topological Ring is Continuous, $f g$ is continuous for each $f, g \in \map {\CC_b} X$.
Further, from Product of Bounded Map... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $\map {\CC_b} {X; \C} := \map {\CC_b} X$ be the [[Definition:Space of Bounded Continuous Functions on Topological Space|space of bounded continuous functions on $X$ valued in $\C$]].
Let $\norm {\,\cdot\,}_\infty$ be the [[Definition:Supremum Norm|... | From [[Bounded Continuous Functions on Topological Space form Banach Space]], $\struct {\map {\CC_b} X, \norm {\,\cdot\,}_\infty}$ is a [[Definition:Banach Space|Banach space]] over $\C$.
From [[Product of Continuous Functions on Topological Ring is Continuous]], $f g$ is [[Definition:Continuous Mapping|continuous]] f... | Bounded Complex-Valued Continuous Functions on Topological Space form Unital C*-Algebra | https://proofwiki.org/wiki/Bounded_Complex-Valued_Continuous_Functions_on_Topological_Space_form_Unital_C*-Algebra | https://proofwiki.org/wiki/Bounded_Complex-Valued_Continuous_Functions_on_Topological_Space_form_Unital_C*-Algebra | [
"C*-Algebras"
] | [
"Definition:Topological Space",
"Definition:Space of Bounded Continuous Functions on Topological Space",
"Definition:Supremum Norm",
"Definition:Pointwise Multiplication",
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra"
] | [
"Bounded Continuous Functions on Topological Space form Banach Space",
"Definition:Banach Space",
"Product of Continuous Functions on Topological Ring is Continuous",
"Definition:Continuous Mapping",
"Product of Bounded Mappings on Normed Algebra is Bounded",
"Definition:Bounded Mapping/Normed Vector Spac... |
proofwiki-22031 | Closed *-Subalgebra of C*-Algebra is C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $B \subseteq A$ be a closed $\ast$-subalgebra of $A$.
Then $\struct {B, \ast, \norm {\, \cdot \,} }$ is a $\text C^\ast$-algebra. | Since $\struct {B, \ast}$ is a $\ast$-subalgebra of $A$, it is in particular a $\ast$-algebra.
From Closed Subspace of Banach Space forms Banach Space, $\struct {B, \ast, \norm {\, \cdot \,} }$ is additionally a Banach space.
We have:
:$\norm {x y} \le \norm x \norm y$
and:
:$\norm {x x^\ast} = \norm x^2$
for each $x \... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $B \subseteq A$ be a [[Definition:Closed Set|closed]] [[Definition:*-Subalgebra|$\ast$-subalgebra]] of $A$.
Then $\struct {B, \ast, \norm {\, \cdot \,} }$ is a [[Definition:C*-Algebra|$\text C^\ast$-algebra]]. | Since $\struct {B, \ast}$ is a [[Definition:*-Subalgebra|$\ast$-subalgebra]] of $A$, it is in particular a [[Definition:*-Algebra|$\ast$-algebra]].
From [[Closed Subspace of Banach Space forms Banach Space]], $\struct {B, \ast, \norm {\, \cdot \,} }$ is additionally a [[Definition:Banach Space|Banach space]].
We have... | Closed *-Subalgebra of C*-Algebra is C*-Algebra | https://proofwiki.org/wiki/Closed_*-Subalgebra_of_C*-Algebra_is_C*-Algebra | https://proofwiki.org/wiki/Closed_*-Subalgebra_of_C*-Algebra_is_C*-Algebra | [
"C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Closed Set",
"Definition:*-Subalgebra",
"Definition:C*-Algebra"
] | [
"Definition:*-Subalgebra",
"Definition:*-Algebra",
"Closed Subspace of Banach Space forms Banach Space",
"Definition:Banach Space",
"Definition:C*-Algebra",
"Category:C*-Algebras"
] |
proofwiki-22032 | Complex Conjugate of Complex-Valued Function Vanishing at Infinity Vanishes at Infinity | Let $X$ be a locally compact Hausdorff space.
Let $f : X \to \C$ be a complex-valued function vanishing at infinity.
Define $\overline f : X \to \C$ by:
:$\forall x \in \C: \map {\overline f} x = \overline {\map f x}$
where $\overline {\map f x}$ denotes the conjugate of $\overline {\map f x}$.
Then $\overline f$ is a... | {{explain|What is the significance of $X$ being a locally compact Hausdorff space? What would happen to this proof if it were not?}}
Let $\epsilon > 0$.
Then since $f$ vanishes at infinity, there exists a compact set $K \subseteq X$ such that:
:$\cmod {\map f x} < \epsilon$ for all $x \in X \setminus K$.
From Complex ... | Let $X$ be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]].
Let $f : X \to \C$ be a [[Definition:Complex-Valued Function Vanishing at Infinity|complex-valued function vanishing at infinity]].
Define $\overline f : X \to \C$ by:
:$\forall x \in \C: \map {\overline f} x = \overline {\m... | {{explain|What is the significance of $X$ being a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]]? What would happen to this proof if it were not?}}
Let $\epsilon > 0$.
Then since $f$ [[Definition:Complex-Valued Function Vanishing at Infinity|vanishes at infinity]], there exists a [[Def... | Complex Conjugate of Complex-Valued Function Vanishing at Infinity Vanishes at Infinity | https://proofwiki.org/wiki/Complex_Conjugate_of_Complex-Valued_Function_Vanishing_at_Infinity_Vanishes_at_Infinity | https://proofwiki.org/wiki/Complex_Conjugate_of_Complex-Valued_Function_Vanishing_at_Infinity_Vanishes_at_Infinity | [
"Complex-Valued Functions Vanishing at Infinity"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Complex Conjugate",
"Definition:Complex-Valued Function Vanishing at Infinity"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Compact Topological Space/Subspace",
"Complex Modulus equals Complex Modulus of Conjugate",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Category:Complex-Valued Functions V... |
proofwiki-22033 | Space of Continuous Functions Vanishing at Infinity is C*-Algebra | Let $X$ be a locally compact Hausdorff space.
Let $\map {\CC_0} X$ be the set of continuous complex-valued functions vanishing at infinity.
Equip $\map {\CC_0} X$ with pointwise addition, pointwise scalar multiplication and pointwise multiplication.
For each $f \in \map {\CC_0} X$, define:
:$\map {\overline f} x = \ov... | From Bounded Complex-Valued Continuous Functions on Topological Space form Unital C*-Algebra, $\tuple {\map {\CC_b} X, \overline {\, \cdot \,}, \norm {\, \cdot \,}_\infty}$ is a $\text C^\ast$-algebra where $\map {\CC_b} X$ is the space of bounded continuous functions on $X$ valued in $\C$.
From Closed *-Subalgebra of ... | Let $X$ be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]].
Let $\map {\CC_0} X$ be the [[Definition:Set|set]] of [[Definition:Continuous Mapping|continuous]] [[Definition:Complex-Valued Function Vanishing at Infinity|complex-valued functions vanishing at infinity]].
Equip $\map {\CC... | From [[Bounded Complex-Valued Continuous Functions on Topological Space form Unital C*-Algebra]], $\tuple {\map {\CC_b} X, \overline {\, \cdot \,}, \norm {\, \cdot \,}_\infty}$ is a [[Definition:C*-Algebra|$\text C^\ast$-algebra]] where $\map {\CC_b} X$ is the [[Definition:Space of Bounded Continuous Functions on Topol... | Space of Continuous Functions Vanishing at Infinity is C*-Algebra | https://proofwiki.org/wiki/Space_of_Continuous_Functions_Vanishing_at_Infinity_is_C*-Algebra | https://proofwiki.org/wiki/Space_of_Continuous_Functions_Vanishing_at_Infinity_is_C*-Algebra | [
"C*-Algebras",
"Complex-Valued Functions Vanishing at Infinity"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:Set",
"Definition:Continuous Mapping",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Pointwise Addition",
"Definition:Pointwise Scalar Multiplication of Mappings",
"Definition:Pointwise Multiplication",
"Definition:Sup... | [
"Bounded Complex-Valued Continuous Functions on Topological Space form Unital C*-Algebra",
"Definition:C*-Algebra",
"Definition:Space of Bounded Continuous Functions on Topological Space",
"Closed *-Subalgebra of C*-Algebra is C*-Algebra",
"Definition:Closed Set",
"Definition:*-Subalgebra",
"Space of Co... |
proofwiki-22034 | Stone-Weierstrass Theorem/Compact Space/Complex-Valued | Let $\map \CC {X, \C}$ be the set of complex-valued continuous functions on $X$.
Equip $\map \CC {X, \C}$ with pointwise addition, pointwise scalar multiplication and pointwise multiplication.
For each $f \in \map \CC {X, \C}$, define $\overline f : X \to \C$ by:
:$\map {\overline f} x = \overline {\map f x}$
for each ... | Let $\BB$ be the set:
:$\BB = \set {f \in \AA : f \in \map \CC {X, \R} }$
We show that $\BB$ is a algebra over $\R$.
From Bounded Continuous Functions on Topological Space form Banach Space, $\BB$ is a vector space over $\R$.
From Product of Continuous Functions on Topological Ring is Continuous, $\BB$ is an algebra ov... | Let $\map \CC {X, \C}$ be the [[Definition:Set|set]] of [[Definition:Complex-Valued Function|complex-valued]] [[Definition:Continuous Mapping|continuous functions]] on $X$.
Equip $\map \CC {X, \C}$ with [[Definition:Pointwise Addition|pointwise addition]], [[Definition:Pointwise Scalar Multiplication|pointwise scalar ... | Let $\BB$ be the [[Definition:Set|set]]:
:$\BB = \set {f \in \AA : f \in \map \CC {X, \R} }$
We show that $\BB$ is a [[Definition:Algebra over Field|algebra over $\R$]].
From [[Bounded Continuous Functions on Topological Space form Banach Space]], $\BB$ is a [[Definition:Vector Space|vector space]] over $\R$.
From [... | Stone-Weierstrass Theorem/Compact Space/Complex-Valued | https://proofwiki.org/wiki/Stone-Weierstrass_Theorem/Compact_Space/Complex-Valued | https://proofwiki.org/wiki/Stone-Weierstrass_Theorem/Compact_Space/Complex-Valued | [
"Stone-Weierstrass Theorem"
] | [
"Definition:Set",
"Definition:Complex-Valued Function",
"Definition:Continuous Mapping",
"Definition:Pointwise Addition",
"Definition:Pointwise Scalar Multiplication of Mappings",
"Definition:Pointwise Multiplication",
"Definition:Unital Subalgebra",
"Definition:*-Subalgebra",
"Definition:Mappings S... | [
"Definition:Set",
"Definition:Algebra over Field",
"Bounded Continuous Functions on Topological Space form Banach Space",
"Definition:Vector Space",
"Product of Continuous Functions on Topological Ring is Continuous",
"Definition:Algebra over Field",
"Definition:Subalgebra",
"Sum of Complex Number wit... |
proofwiki-22035 | Stone-Weierstrass Theorem/Locally Compact Hausdorff Space | Let $X$ be a locally compact Hausdorff space.
Let $\struct {\map {\CC_0} {X, \R}, \norm {\, \cdot \,} }$ be the Banach algebra of real-valued continuous functions vanishing at infinity on $X$.
Let $\AA$ be a subalgebra of $\map {\CC_0} {X, \R}$ such that:
:$(1) \quad$ for each $x, y \in X$ with $x \ne y$ there exists $... | Let $X^\ast = X \cup \set p$ be the Alexandroff extension of $X$.
From Extension of Continuous Complex-Valued Function Vanishing at Infinity to Alexandroff Extension is Continuous, for each $f \in \map {\CC_0} {X, \R}$ we can define:
:$\map {f^\ast} x = \begin{cases}\map f x & x \in X \\ 0 & x = p\end{cases}$
for each... | Let $X$ be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]].
Let $\struct {\map {\CC_0} {X, \R}, \norm {\, \cdot \,} }$ be the [[Definition:Banach Algebra|Banach algebra]] of [[Definition:Real-Valued Function|real-valued]] [[Definition:Complex-Valued Function Vanishing at Infinity|conti... | Let $X^\ast = X \cup \set p$ be the [[Definition:Alexandroff Extension|Alexandroff extension]] of $X$.
From [[Extension of Continuous Complex-Valued Function Vanishing at Infinity to Alexandroff Extension is Continuous]], for each $f \in \map {\CC_0} {X, \R}$ we can define:
:$\map {f^\ast} x = \begin{cases}\map f x &... | Stone-Weierstrass Theorem/Locally Compact Hausdorff Space | https://proofwiki.org/wiki/Stone-Weierstrass_Theorem/Locally_Compact_Hausdorff_Space | https://proofwiki.org/wiki/Stone-Weierstrass_Theorem/Locally_Compact_Hausdorff_Space | [
"Stone-Weierstrass Theorem"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:Banach Algebra",
"Definition:Real-Valued Function",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Subalgebra",
"Definition:Everywhere Dense"
] | [
"Definition:Alexandroff Extension",
"Extension of Continuous Complex-Valued Function Vanishing at Infinity to Alexandroff Extension is Continuous",
"Alexandroff Extension is Compact",
"Definition:Compact Topological Space",
"Constant Mapping is Continuous",
"Definition:Subalgebra",
"Definition:Algebra o... |
proofwiki-22036 | Stone-Weierstrass Theorem/Locally Compact Hausdorff Space/Complex-Valued | Let $X$ be a locally compact Hausdorff space.
Let $\struct {\map {\CC_0} {X, \C}, \norm {\, \cdot \,} }$ be the Banach algebra of complex-valued continuous functions vanishing at infinity on $X$.
Let $\AA$ be a $\ast$-subalgebra of $\map {\CC_0} {X, \C}$ such that:
:$(1) \quad$ for each $x, y \in X$ with $x \ne y$ ther... | This proof is essentially identical to Stone-Weierstrass Theorem: Compact Space: Complex-Valued.
We define:
:$\BB = \set {\map \Re f : f \in \AA} = \set {\map \Im f : f \in \AA}$
Replicating the proof in Stone-Weierstrass Theorem: Compact Space: Complex-Valued, it can be shown:
:$\BB = \set {\map \Re f : f \in \AA} = \... | Let $X$ be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]].
Let $\struct {\map {\CC_0} {X, \C}, \norm {\, \cdot \,} }$ be the [[Definition:Banach Algebra|Banach algebra]] of [[Definition:Complex-Valued Function Vanishing at Infinity|complex-valued continuous functions vanishing at infi... | This proof is essentially identical to [[Stone-Weierstrass Theorem/Compact Space/Complex-Valued|Stone-Weierstrass Theorem: Compact Space: Complex-Valued]].
We define:
:$\BB = \set {\map \Re f : f \in \AA} = \set {\map \Im f : f \in \AA}$
Replicating the proof in [[Stone-Weierstrass Theorem/Compact Space/Complex-Value... | Stone-Weierstrass Theorem/Locally Compact Hausdorff Space/Complex-Valued | https://proofwiki.org/wiki/Stone-Weierstrass_Theorem/Locally_Compact_Hausdorff_Space/Complex-Valued | https://proofwiki.org/wiki/Stone-Weierstrass_Theorem/Locally_Compact_Hausdorff_Space/Complex-Valued | [
"Stone-Weierstrass Theorem"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:Banach Algebra",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:*-Subalgebra",
"Definition:Everywhere Dense"
] | [
"Stone-Weierstrass Theorem/Compact Space/Complex-Valued",
"Stone-Weierstrass Theorem/Compact Space/Complex-Valued",
"Definition:Mappings Separating Points",
"Stone-Weierstrass Theorem/Locally Compact Hausdorff Space",
"Real and Imaginary Parts of Complex-Valued Function Vanishing at Infinity Vanish at Infin... |
proofwiki-22037 | Real and Imaginary Parts of Complex-Valued Function Vanishing at Infinity Vanish at Infinity | Let $X$ be a locally compact Hausdorff space.
Let $f : X \to \C$ be a complex-valued function vanishing at infinity.
Then $\map \Re f$ and $\map \Im f$ vanish at infinity. | Let $\epsilon > 0$.
Since $f$ vanishes at infinity, there exists a compact $K \subseteq X$ such that:
:$\cmod {\map f x}^2 < \epsilon^2$ for each $x \in X \setminus K$.
Then we have:
:$\paren {\map \Re {\map f x} }^2 + \paren {\map \Im {\map f x} }^2 < \epsilon^2$
So that:
:$\paren {\map \Re {\map f x} }^2 < \epsilo... | Let $X$ be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]].
Let $f : X \to \C$ be a [[Definition:Complex-Valued Function Vanishing at Infinity|complex-valued function vanishing at infinity]].
Then $\map \Re f$ and $\map \Im f$ [[Definition:Complex-Valued Function Vanishing at Infinit... | Let $\epsilon > 0$.
Since $f$ [[Definition:Complex-Valued Function Vanishing at Infinity|vanishes at infinity]], there exists a [[Definition:Compact Topological Space|compact]] $K \subseteq X$ such that:
:$\cmod {\map f x}^2 < \epsilon^2$ for each $x \in X \setminus K$.
Then we have:
:$\paren {\map \Re {\map f x} }... | Real and Imaginary Parts of Complex-Valued Function Vanishing at Infinity Vanish at Infinity | https://proofwiki.org/wiki/Real_and_Imaginary_Parts_of_Complex-Valued_Function_Vanishing_at_Infinity_Vanish_at_Infinity | https://proofwiki.org/wiki/Real_and_Imaginary_Parts_of_Complex-Valued_Function_Vanishing_at_Infinity_Vanish_at_Infinity | [
"Complex-Valued Functions Vanishing at Infinity"
] | [
"Definition:Locally Compact Hausdorff 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-22038 | Newton's Three-Eighths Rule | Let $f$ be a real function which is integrable on the closed interval $\closedint a b$.
Let $P = \set {a = x_0, x_1, x_2, \ldots, x_{3 n - 1}, x_{3 n} = b}$ form a normal subdivision of $\closedint a b$:
:$\forall r \in \set {1, 2, \ldots, 3 n}: x_r - x_{r - 1} = \dfrac {b - a} {3 n}$
Then the definite integral of $f$ ... | {{ProofWanted|Graphical approach}} | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Integrable Function|integrable]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Let $P = \set {a = x_0, x_1, x_2, \ldots, x_{3 n - 1}, x_{3 n} = b}$ form a [[Definition:Normal Subdivision|normal subdivision]] of... | {{ProofWanted|Graphical approach}} | Newton's Three-Eighths Rule | https://proofwiki.org/wiki/Newton's_Three-Eighths_Rule | https://proofwiki.org/wiki/Newton's_Three-Eighths_Rule | [
"Newton's Three-Eighths Rule",
"Definite Integrals"
] | [
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Real Interval/Closed",
"Definition:Subdivision of Interval/Normal Subdivision",
"Definition:Definite Integral",
"Definition:Approximation"
] | [] |
proofwiki-22039 | Spectrum of Element of Unital Commutative Banach Algebra/Corollary 1 | Let $\struct {A, \norm {\, \cdot \,} }$ be a commutative Banach algebra over $\C$ that is not unital as an algebra. | Let $\struct {A_+, \norm {\, \cdot \,}_{A_+} }$ be the unitization of $\struct {A, \norm {\, \cdot \,} }$.
By the definition of the spectrum, we have:
:$\map {\sigma_A} x = \map {\sigma_{A_+} } {\tuple {x, 0} }$
Let $\Phi_{A_+}$ be the spectrum of $A_+$.
From Unitization of Commutative Algebra over Field is Commutative... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Commutative Algebra (Abstract Algebra)|commutative]] [[Definition:Banach Algebra|Banach algebra]] over $\C$ that is not [[Definition:Unital Algebra|unital as an algebra]]. | Let $\struct {A_+, \norm {\, \cdot \,}_{A_+} }$ be the [[Definition:Unitization of Normed Algebra|unitization]] of $\struct {A, \norm {\, \cdot \,} }$.
By the definition of the [[Definition:Spectrum (Spectral Theory)|spectrum]], we have:
:$\map {\sigma_A} x = \map {\sigma_{A_+} } {\tuple {x, 0} }$
Let $\Phi_{A_+}$ be... | Spectrum of Element of Unital Commutative Banach Algebra/Corollary 1 | https://proofwiki.org/wiki/Spectrum_of_Element_of_Unital_Commutative_Banach_Algebra/Corollary_1 | https://proofwiki.org/wiki/Spectrum_of_Element_of_Unital_Commutative_Banach_Algebra/Corollary_1 | [
"Spectrum of Element of Unital Commutative Banach Algebra"
] | [
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Banach Algebra",
"Definition:Unital Algebra"
] | [
"Definition:Unitization of Normed Algebra",
"Definition:Spectrum (Spectral Theory)",
"Definition:Spectrum of Banach Algebra",
"Unitization of Commutative Algebra over Field is Commutative",
"Definition:Commutative Algebra (Abstract Algebra)",
"Spectrum of Element of Unital Commutative Banach Algebra",
"... |
proofwiki-22040 | Existence of Maximal Commutative Subalgebra/Unital | Let $K$ be a field.
Let $A$ be an unital algebra over $K$.
Let $S \subseteq A$ be a non-empty set such that:
:for all $x, y \in S$ we have $x y = y x$.
Then there exists a commutative unital subalgebra of $A$ containing $S$ that is maximal (among all subalgebras) {{WRT}} set inclusion. | Let ${\mathbf 1}_A$ be the identity element of $A$.
Note that ${\mathbf 1}_A$ commutes with every element of $S$.
Hence $S' = S \cup \set { {\mathbf 1}_A}$ is such that:
:for all $x, y \in S'$ we have $x y = y x$.
Hence applying Existence of Maximal Commutative Subalgebra to $S'$, there exists a commutative subalgebr... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $A$ be an [[Definition:Unital Algebra|unital algebra]] over $K$.
Let $S \subseteq A$ be a [[Definition:Non-Empty Set|non-empty set]] such that:
:for all $x, y \in S$ we have $x y = y x$.
Then there exists a [[Definition:Commutative Algebra (Abstract ... | Let ${\mathbf 1}_A$ be the [[Definition:Identity Element|identity element]] of $A$.
Note that ${\mathbf 1}_A$ [[Definition:Commutative/Set|commutes]] with every element of $S$.
Hence $S' = S \cup \set { {\mathbf 1}_A}$ is such that:
:for all $x, y \in S'$ we have $x y = y x$.
Hence applying [[Existence of Maximal ... | Existence of Maximal Commutative Subalgebra/Unital | https://proofwiki.org/wiki/Existence_of_Maximal_Commutative_Subalgebra/Unital | https://proofwiki.org/wiki/Existence_of_Maximal_Commutative_Subalgebra/Unital | [
"Existence of Maximal Commutative Subalgebra"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Unital Algebra",
"Definition:Non-Empty Set",
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Unital Subalgebra",
"Definition:Maximal/Element",
"Definition:Subalgebra",
"Definition:Subset"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Commutative/Set",
"Existence of Maximal Commutative Subalgebra",
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Subalgebra",
"Definition:Maximal/Element",
"Definition:Subalgebra",
"Definition:Subset",
"Definit... |
proofwiki-22041 | Gelfand Transform is Continuous Function Vanishing at Infinity | Let $\struct {A, \norm {\, \cdot \,} }$ be a commutative Banach algebra over $\C$ such that:
:the spectrum $\Phi_A$ of $A$ is non-empty.
Let $\hat a$ be the Gelfand transform of $a$.
Let $\struct {\map {\CC_0} {\Phi_A}, \norm {\, \cdot \,}_\infty}$ be the set of complex-valued functions vanishing at infinity with the ... | From Characterization of Continuity of Linear Functional in Weak-* Topology:
:the map $\iota a : \struct {A^{\ast \ast}, w^\ast} \to \C$ defined by $\map {\iota a} \phi = \map \phi a$ for each $\phi \in A^{\ast \ast}$ is continuous.
The spectrum $\Phi_A$ is given the subspace topology inherited from $\struct {A^{\ast ... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Commutative Algebra (Abstract Algebra)|commutative]] [[Definition:Banach Algebra|Banach algebra]] over $\C$ such that:
:the [[Definition:Spectrum of Banach Algebra|spectrum]] $\Phi_A$ of $A$ is [[Definition:Non-Empty Set|non-empty]].
Let $\hat a$ be the [[Defi... | From [[Characterization of Continuity of Linear Functional in Weak-* Topology]]:
:the map $\iota a : \struct {A^{\ast \ast}, w^\ast} \to \C$ defined by $\map {\iota a} \phi = \map \phi a$ for each $\phi \in A^{\ast \ast}$ is [[Definition:Continuous Mapping|continuous]].
The [[Definition:Spectrum of Banach Algebra|spe... | Gelfand Transform is Continuous Function Vanishing at Infinity | https://proofwiki.org/wiki/Gelfand_Transform_is_Continuous_Function_Vanishing_at_Infinity | https://proofwiki.org/wiki/Gelfand_Transform_is_Continuous_Function_Vanishing_at_Infinity | [
"Gelfand Transforms",
"Commutative Banach Algebras"
] | [
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Banach Algebra",
"Definition:Spectrum of Banach Algebra",
"Definition:Non-Empty Set",
"Definition:Gelfand Transform",
"Definition:Set",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Supremum Norm"
] | [
"Characterization of Continuity of Linear Functional in Weak-* Topology",
"Definition:Continuous Mapping",
"Definition:Spectrum of Banach Algebra",
"Definition:Topological Subspace",
"Definition:Continuous Mapping",
"Restriction of Continuous Mapping is Continuous/Topological Spaces",
"Definition:Comple... |
proofwiki-22042 | Gelfand Representation Theorem | Let $\struct {A, \norm {\, \cdot \,} }$ be a commutative Banach algebra over $\C$ such that:
:the spectrum $\Phi_A$ of $A$ is non-empty.
Let $\hat a$ be the Gelfand transform of $a$.
Let $\struct {\map {\CC_0} {\Phi_A}, \norm {\, \cdot \,}_\infty}$ be the set of complex-valued functions vanishing at infinity with the ... | === Proof that $G$ is an algebra homomorphism ===
Let $a, b \in A$, $\lambda \in \C$ and $\phi \in \Phi_A$.
Then we have:
{{begin-eqn}}
{{eqn | l = \map {\paren {\map G {a + \lambda b} } } \phi
| r = \map {\widehat {a + \lambda b} } \phi
}}
{{eqn | r = \map \phi {a + \lambda b}
}}
{{eqn | r = \map \phi a + \lambda ... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Commutative Algebra (Abstract Algebra)|commutative]] [[Definition:Banach Algebra|Banach algebra]] over $\C$ such that:
:the [[Definition:Spectrum of Banach Algebra|spectrum]] $\Phi_A$ of $A$ is [[Definition:Non-Empty Set|non-empty]].
Let $\hat a$ be the [[Defi... | === Proof that $G$ is an [[Definition:Algebra Homomorphism|algebra homomorphism]] ===
Let $a, b \in A$, $\lambda \in \C$ and $\phi \in \Phi_A$.
Then we have:
{{begin-eqn}}
{{eqn | l = \map {\paren {\map G {a + \lambda b} } } \phi
| r = \map {\widehat {a + \lambda b} } \phi
}}
{{eqn | r = \map \phi {a + \lambda b}... | Gelfand Representation Theorem | https://proofwiki.org/wiki/Gelfand_Representation_Theorem | https://proofwiki.org/wiki/Gelfand_Representation_Theorem | [
"Gelfand Transforms",
"Commutative Banach Algebras"
] | [
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Banach Algebra",
"Definition:Spectrum of Banach Algebra",
"Definition:Non-Empty Set",
"Definition:Gelfand Transform",
"Definition:Set",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Supremum Norm",
"Definition... | [
"Definition:Algebra Homomorphism",
"Definition:Linear Functional",
"Definition:Linear Transformation",
"Definition:Algebra Homomorphism"
] |
proofwiki-22043 | Spectrum of Element of Unital Commutative Banach Algebra/Corollary 2 | Let $\struct {A, \norm {\, \cdot \,} }$ be a commutative unital Banach algebra over $\C$.
Let $\Phi_A$ be the spectrum of $A$.
Then $\Phi_A \ne \O$. | From Spectrum of Element of Banach Algebra is Non-Empty, we have $\map {\sigma_A} x \ne \O$.
That is, there exists $\lambda \in \map {\sigma_A} x$.
From Spectrum of Element of Unital Commutative Banach Algebra, there exists a character $\phi$ with $\map \phi x = \lambda$.
In particular, $\phi \in \Phi_A$.
So $\Phi_A \n... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Commutative Algebra (Abstract Algebra)|commutative]] [[Definition:Unital Banach Algebra|unital Banach algebra]] over $\C$.
Let $\Phi_A$ be the [[Definition:Spectrum of Banach Algebra|spectrum]] of $A$.
Then $\Phi_A \ne \O$. | From [[Spectrum of Element of Banach Algebra is Non-Empty]], we have $\map {\sigma_A} x \ne \O$.
That is, there exists $\lambda \in \map {\sigma_A} x$.
From [[Spectrum of Element of Unital Commutative Banach Algebra]], there exists a [[Definition:Character (Banach Algebra)|character]] $\phi$ with $\map \phi x = \lamb... | Spectrum of Element of Unital Commutative Banach Algebra/Corollary 2 | https://proofwiki.org/wiki/Spectrum_of_Element_of_Unital_Commutative_Banach_Algebra/Corollary_2 | https://proofwiki.org/wiki/Spectrum_of_Element_of_Unital_Commutative_Banach_Algebra/Corollary_2 | [
"Spectrum of Element of Unital Commutative Banach Algebra"
] | [
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Unital Banach Algebra",
"Definition:Spectrum of Banach Algebra"
] | [
"Spectrum of Element of Banach Algebra is Non-Empty",
"Spectrum of Element of Unital Commutative Banach Algebra",
"Definition:Character (Banach Algebra)",
"Category:Spectrum of Element of Unital Commutative Banach Algebra"
] |
proofwiki-22044 | C*-Algebra embeds into Multiplier Algebra as C*-Subalgebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $\struct {\map M A, \ast, \norm {\, \cdot \,}_{\map M A} }$ be the multiplier algebra of $A$.
Define $\phi : A \to \map M A$ by:
:$\map \phi a = \tuple {L_a, R_a}$
for each $a \in A$, where $\tuple {L_a, R_a}$ is the double centralizer gener... | We show that $\phi$ is a linear isometry.
Let $a, b \in A$ and $\lambda \in \C$.
Then we have for $c \in A$:
{{begin-eqn}}
{{eqn | l = \map {L_{a + \lambda b} } c
| r = \paren {a + \lambda b} c
}}
{{eqn | r = a c + \lambda b c
}}
{{eqn | r = \map {L_a} c + \lambda \map {L_b} c
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn ... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\struct {\map M A, \ast, \norm {\, \cdot \,}_{\map M A} }$ be the [[Definition:Multiplier Algebra|multiplier algebra]] of $A$.
Define $\phi : A \to \map M A$ by:
:$\map \phi a = \tuple {L_a, R_a}$
for each $a \in... | We show that $\phi$ is a [[Definition:Linear|linear isometry]].
Let $a, b \in A$ and $\lambda \in \C$.
Then we have for $c \in A$:
{{begin-eqn}}
{{eqn | l = \map {L_{a + \lambda b} } c
| r = \paren {a + \lambda b} c
}}
{{eqn | r = a c + \lambda b c
}}
{{eqn | r = \map {L_a} c + \lambda \map {L_b} c
}}
{{end-eqn}}
a... | C*-Algebra embeds into Multiplier Algebra as C*-Subalgebra | https://proofwiki.org/wiki/C*-Algebra_embeds_into_Multiplier_Algebra_as_C*-Subalgebra | https://proofwiki.org/wiki/C*-Algebra_embeds_into_Multiplier_Algebra_as_C*-Subalgebra | [
"Multiplier Algebras",
"C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Multiplier Algebra",
"Definition:Double Centralizer Generated by Element of C*-Algebra",
"Definition:C*-Subalgebra"
] | [
"Definition:Linear",
"Definition:Linear Transformation",
"Norm of Double Centralizer Generated by Element of C*-Algebra",
"Definition:Linear Isometry",
"Definition:Algebra Homomorphism",
"Definition:Algebra Homomorphism",
"Definition:*-Algebra Homomorphism",
"Image of *-Algebra under *-Algebra Homomor... |
proofwiki-22045 | Image of Algebra under Algebra Homomorphism is Subalgebra | Let $R$ be a ring.
Let $A, B$ be $R$-algebras.
Let $\phi : A \to B$ be an $R$-algebra homomorphism.
Then $\phi \sqbrk A$ is a subalgebra of $B$. | From Image of Submodule under Linear Transformation is Submodule, $\phi \sqbrk A$ is a submodule of $B$.
Let $a, b \in A$.
We then have:
:$\map \phi a \map \phi b = \map \phi {a b} \in \phi \sqbrk A$
since $\phi$ is a $R$-algebra homomorphism.
Hence $\phi \sqbrk A$ is a subalgebra of $B$.
{{qed}}
Category:Algebras
3... | Let $R$ be a [[Definition:Ring|ring]].
Let $A, B$ be [[Definition:Algebra over Ring|$R$-algebras]].
Let $\phi : A \to B$ be an [[Definition:Algebra Homomorphism|$R$-algebra homomorphism]].
Then $\phi \sqbrk A$ is a [[Definition:Subalgebra|subalgebra]] of $B$. | From [[Image of Submodule under Linear Transformation is Submodule]], $\phi \sqbrk A$ is a [[Definition:Submodule|submodule]] of $B$.
Let $a, b \in A$.
We then have:
:$\map \phi a \map \phi b = \map \phi {a b} \in \phi \sqbrk A$
since $\phi$ is a [[Definition:Algebra Homomorphism|$R$-algebra homomorphism]].
Hence... | Image of Algebra under Algebra Homomorphism is Subalgebra | https://proofwiki.org/wiki/Image_of_Algebra_under_Algebra_Homomorphism_is_Subalgebra | https://proofwiki.org/wiki/Image_of_Algebra_under_Algebra_Homomorphism_is_Subalgebra | [
"Algebras"
] | [
"Definition:Ring",
"Definition:Algebra over Ring",
"Definition:Algebra Homomorphism",
"Definition:Subalgebra"
] | [
"Image of Submodule under Linear Transformation is Submodule",
"Definition:Submodule",
"Definition:Algebra Homomorphism",
"Definition:Subalgebra",
"Category:Algebras"
] |
proofwiki-22046 | Image of *-Algebra under *-Algebra Homomorphism is *-Subalgebra | Let $\struct {A, \ast}$ and $\struct {B, \square}$ be a $\ast$-algebras over $\C$.
Let $\phi : A \to B$ be a $\ast$-algebra homomorphism.
Then $\phi \sqbrk A$ is a $\ast$-subalgebra of $B$. | From Image of Algebra under Algebra Homomorphism is Subalgebra, $\phi \sqbrk A$ is a subalgebra of $B$.
Further, for $a \in A$ we have:
:$\paren {\map \phi a}^\square = \map \phi {a^\ast} \in \phi \sqbrk A$
So:
:$x^\ast \in \phi \sqbrk A$ for $x \in \phi \sqbrk A$.
Hence $\phi \sqbrk A$ is a $\ast$-subalgebra of $B$.
... | Let $\struct {A, \ast}$ and $\struct {B, \square}$ be a [[Definition:*-Algebra|$\ast$-algebras]] over $\C$.
Let $\phi : A \to B$ be a [[Definition:*-Algebra Homomorphism|$\ast$-algebra homomorphism]].
Then $\phi \sqbrk A$ is a [[Definition:*-Subalgebra|$\ast$-subalgebra]] of $B$. | From [[Image of Algebra under Algebra Homomorphism is Subalgebra]], $\phi \sqbrk A$ is a [[Definition:Subalgebra|subalgebra]] of $B$.
Further, for $a \in A$ we have:
:$\paren {\map \phi a}^\square = \map \phi {a^\ast} \in \phi \sqbrk A$
So:
:$x^\ast \in \phi \sqbrk A$ for $x \in \phi \sqbrk A$.
Hence $\phi \sqbrk A... | Image of *-Algebra under *-Algebra Homomorphism is *-Subalgebra | https://proofwiki.org/wiki/Image_of_*-Algebra_under_*-Algebra_Homomorphism_is_*-Subalgebra | https://proofwiki.org/wiki/Image_of_*-Algebra_under_*-Algebra_Homomorphism_is_*-Subalgebra | [
"*-Algebras"
] | [
"Definition:*-Algebra",
"Definition:*-Algebra Homomorphism",
"Definition:*-Subalgebra"
] | [
"Image of Algebra under Algebra Homomorphism is Subalgebra",
"Definition:Subalgebra",
"Definition:*-Subalgebra",
"Category:*-Algebras"
] |
proofwiki-22047 | Image of Closed Set under Linear Isometry from Banach Space to Normed Vector Space is Closed | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $\struct {Y, \norm {\, \cdot \,}_Y}$ be a normed vector space over $\GF$.
Let $T : X \to Y$ be a linear isometry.
Let $A$ be a closed subset of $X$.
Then $T \sqbrk A$ is closed in $X$. | Let $\sequence {y_n}_{n \mathop \in \N}$ be a convergent sequence in $T \sqbrk A$ with:
:$y_n \to y$ as $n \to \infty$.
From Convergent Sequence is Cauchy Sequence, $\sequence {y_n}_{n \mathop \in \N}$ is a Cauchy sequence.
Then there exists $x_n \in X$ with $y_n = T x_n$.
Let $\epsilon > 0$.
Let $N \in \N$ be such t... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $\struct {Y, \norm {\, \cdot \,}_Y}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $T : X \to Y$ be a [[Definition:Linear Isometry|linear isometry]].
L... | Let $\sequence {y_n}_{n \mathop \in \N}$ be a [[Definition:Convergent Sequence|convergent sequence]] in $T \sqbrk A$ with:
:$y_n \to y$ as $n \to \infty$.
From [[Convergent Sequence is Cauchy Sequence]], $\sequence {y_n}_{n \mathop \in \N}$ is a [[Definition:Cauchy Sequence|Cauchy sequence]].
Then there exists $x_n... | Image of Closed Set under Linear Isometry from Banach Space to Normed Vector Space is Closed | https://proofwiki.org/wiki/Image_of_Closed_Set_under_Linear_Isometry_from_Banach_Space_to_Normed_Vector_Space_is_Closed | https://proofwiki.org/wiki/Image_of_Closed_Set_under_Linear_Isometry_from_Banach_Space_to_Normed_Vector_Space_is_Closed | [
"Linear Isometries",
"Banach Spaces"
] | [
"Definition:Banach Space",
"Definition:Normed Vector Space",
"Definition:Linear Isometry",
"Definition:Closed Set",
"Definition:Closed Set"
] | [
"Definition:Convergent Sequence",
"Convergent Sequence is Cauchy Sequence",
"Definition:Cauchy Sequence",
"Definition:Linear Isometry",
"Definition:Cauchy Sequence",
"Definition:Closed Set",
"Subset of Metric Space contains Limits of Sequences iff Closed",
"Definition:Continuous Mapping",
"Limit in ... |
proofwiki-22048 | Existence of Unitization of C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra that is not unital as an algebra.
Let $A_+$ be the unitization of $A$ as an algebra.
Define:
:$\tuple {x, \lambda}^\ast = \tuple {x^\ast, \overline \lambda}$
for each $\tuple {x, \lambda} \in A_+$.
Then there exists a unique algebra norm $\norm {... | Let $\map M A$ be the multiplier algebra of $A$.
Define $T : A_+ \to \map M A$ by:
:$\map T {a, \lambda} = \tuple {L_a + \lambda I_A, R_a + \lambda I_A}$
where $\tuple {L_a, R_a}$ is the double centralizer generated by $a$ and $I_A$ is the identity map.
We show that $T$ is an $\ast$-algebra homomorphism so that $T \sq... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]] that is not [[Definition:Unital Algebra|unital as an algebra]].
Let $A_+$ be the [[Definition:Unitization of Algebra over Field|unitization of $A$ as an algebra]].
Define:
:$\tuple {x, \lambda}^\ast = \tuple {x^\ast, \... | Let $\map M A$ be the [[Definition:Multiplier Algebra|multiplier algebra]] of $A$.
Define $T : A_+ \to \map M A$ by:
:$\map T {a, \lambda} = \tuple {L_a + \lambda I_A, R_a + \lambda I_A}$
where $\tuple {L_a, R_a}$ is the [[Definition:Double Centralizer Generated by Element of C*-Algebra|double centralizer generated b... | Existence of Unitization of C*-Algebra | https://proofwiki.org/wiki/Existence_of_Unitization_of_C*-Algebra | https://proofwiki.org/wiki/Existence_of_Unitization_of_C*-Algebra | [
"Multiplier Algebras",
"C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Unital Algebra",
"Definition:Unitization of Algebra over Field",
"Definition:Norm/Algebra",
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Extension of Mapping"
] | [
"Definition:Multiplier Algebra",
"Definition:Double Centralizer Generated by Element of C*-Algebra",
"Definition:Identity Mapping",
"Definition:*-Algebra Homomorphism",
"Definition:*-Subalgebra",
"C*-Algebra embeds into Multiplier Algebra as C*-Subalgebra",
"Definition:Algebra Homomorphism",
"Definiti... |
proofwiki-22049 | Locale of Open Sets Functor is Covariant | Let $\mathbf{Top}$ denote the category of topological spaces.
Let $\mathbf{Loc}$ denote the category of locales.
Then the open sets functor $\mathbf \Omega : \mathbf{Top} \to \mathbf{Loc}$ is a covariant functor. | Recall the open sets functor $\mathbf \Omega : \mathbf{Top} \to \mathbf{Loc}$ is defined by:
{{DefineFunctor
|ob = $\map \Omega T := $ the locale of topological space $T$
|mor = $\map \Omega f := $ the continuous map induced by continuous mapping $f$
}} | Let $\mathbf{Top}$ denote the [[Definition:Category of Topological Spaces|category of topological spaces]].
Let $\mathbf{Loc}$ denote the [[Definition:Category of Locales|category of locales]].
Then the [[Definition:Locale of Open Sets Functor|open sets functor]] $\mathbf \Omega : \mathbf{Top} \to \mathbf{Loc}$ is a... | Recall the [[Definition:Locale of Open Sets Functor|open sets functor]] $\mathbf \Omega : \mathbf{Top} \to \mathbf{Loc}$ is defined by:
{{DefineFunctor
|ob = $\map \Omega T := $ the [[Definition:Locale of Topological Space|locale of topological space $T$]]
|mor = $\map \Omega f := $ the [[Definition:Continuous Map Ind... | Locale of Open Sets Functor is Covariant | https://proofwiki.org/wiki/Locale_of_Open_Sets_Functor_is_Covariant | https://proofwiki.org/wiki/Locale_of_Open_Sets_Functor_is_Covariant | [
"Functors"
] | [
"Definition:Category of Topological Spaces",
"Definition:Category of Locales",
"Definition:Locale of Open Sets Functor",
"Definition:Functor/Covariant"
] | [
"Definition:Locale of Open Sets Functor",
"Definition:Locale of Topological Space",
"Definition:Continuous Map Induced by Continuous Mapping",
"Definition:Continuous Map Induced by Continuous Mapping"
] |
proofwiki-22050 | Pedal Circle of Incenter is Incircle | Let $\triangle ABC$ be a triangle whose incenter is $H$.
Then the pedal circle of $H$ is the incircle of $\triangle ABC$. | Let $\CC$ denote the incircle of $\triangle ABC$.
{{AimForCont}} $\CC$ is not the pedal circle of $H$.
By definition, the sides of $\triangle ABC$ are tangent to $\CC$.
Hence from {{Porism|Line at Right Angles to Diameter of Circle}}, the line through $H$ to those point of tangency to $\CC$ are perpendicular to the sid... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose [[Definition:Incenter of Triangle|incenter]] is $H$.
Then the [[Definition:Pedal Circle|pedal circle]] of $H$ is the [[Definition:Incircle of Triangle|incircle]] of $\triangle ABC$. | Let $\CC$ denote the [[Definition:Incircle of Triangle|incircle]] of $\triangle ABC$.
{{AimForCont}} $\CC$ is not the [[Definition:Pedal Circle|pedal circle]] of $H$.
By definition, the [[Definition:Side of Polygon|sides]] of $\triangle ABC$ are [[Definition:Tangent Line|tangent]] to $\CC$.
Hence from {{Porism|Line ... | Pedal Circle of Incenter is Incircle | https://proofwiki.org/wiki/Pedal_Circle_of_Incenter_is_Incircle | https://proofwiki.org/wiki/Pedal_Circle_of_Incenter_is_Incircle | [
"Pedal Circles",
"Incenters of Triangles",
"Incircles of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Incircle of Triangle/Incenter",
"Definition:Pedal Circle",
"Definition:Incircle of Triangle"
] | [
"Definition:Incircle of Triangle",
"Definition:Pedal Circle",
"Definition:Polygon/Side",
"Definition:Tangent Line",
"Definition:Line/Straight Line",
"Definition:Points",
"Definition:Tangent Line",
"Definition:Right Angle/Perpendicular",
"Definition:Polygon/Side",
"Definition:Pedal Circle",
"Proo... |
proofwiki-22051 | Pedal Circle of Excenter is Excircle | Let $\triangle ABC$ be a triangle.
Let $H$ be an excenter of $\triangle ABC$.
Then the pedal circle of $H$ is the excircle of $\triangle ABC$ whose center is $H$. | Let $\EE$ denote the excircle of $\triangle ABC$ whose center is $H$.
{{AimForCont}} $\EE$ is not the pedal circle of $H$.
By definition, the sides of $\triangle ABC$ are tangent to $\EE$.
Hence from {{Porism|Line at Right Angles to Diameter of Circle}}, the line through $H$ to those point of tangency to $\EE$ are perp... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $H$ be an [[Definition:Excenter of Triangle|excenter]] of $\triangle ABC$.
Then the [[Definition:Pedal Circle|pedal circle]] of $H$ is the [[Definition:Excircle of Triangle|excircle]] of $\triangle ABC$ whose [[Definition:Center of Circle|cent... | Let $\EE$ denote the [[Definition:Excircle of Triangle|excircle]] of $\triangle ABC$ whose [[Definition:Center of Circle|center]] is $H$.
{{AimForCont}} $\EE$ is not the [[Definition:Pedal Circle|pedal circle]] of $H$.
By definition, the [[Definition:Side of Polygon|sides]] of $\triangle ABC$ are [[Definition:Tangent... | Pedal Circle of Excenter is Excircle | https://proofwiki.org/wiki/Pedal_Circle_of_Excenter_is_Excircle | https://proofwiki.org/wiki/Pedal_Circle_of_Excenter_is_Excircle | [
"Pedal Circles",
"Excenters of Triangles",
"Excircles of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Excircle of Triangle/Excenter",
"Definition:Pedal Circle",
"Definition:Excircle of Triangle",
"Definition:Circle/Center"
] | [
"Definition:Excircle of Triangle",
"Definition:Circle/Center",
"Definition:Pedal Circle",
"Definition:Polygon/Side",
"Definition:Tangent Line",
"Definition:Line/Straight Line",
"Definition:Points",
"Definition:Tangent Line",
"Definition:Right Angle/Perpendicular",
"Definition:Polygon/Side",
"Def... |
proofwiki-22052 | Non-Zero C*-Algebra contains Non-Zero Hermitian Element | Let $\tuple {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$ algebra such that:
:$A \ne \set { {\mathbf 0}_A}$
where ${\mathbf 0}_A$ is the zero vector of $A$.
Then there exists a Hermitian element $x \in A \setminus \set { {\mathbf 0}_A}$. | Let $a \in A \setminus \set { {\mathbf 0}_A}$.
From Product of Element in *-Star Algebra with its Star is Hermitian, $a a^\ast$ is Hermitian.
We just need to show that $a a^\ast \ne {\mathbf 0}_A$.
By the $\text C^\ast$ identity, we have:
:$\norm {a a^\ast} = \norm a^2 \ne 0$
from {{NormAxiomVector|1}}, since $a \ne {... | Let $\tuple {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$ algebra]] such that:
:$A \ne \set { {\mathbf 0}_A}$
where ${\mathbf 0}_A$ is the [[Definition:Zero Vector|zero vector]] of $A$.
Then there exists a [[Definition:Hermitian Element of *-Algebra|Hermitian element]] $x \in A \setminu... | Let $a \in A \setminus \set { {\mathbf 0}_A}$.
From [[Product of Element in *-Star Algebra with its Star is Hermitian]], $a a^\ast$ is [[Definition:Hermitian Element of *-Algebra|Hermitian]].
We just need to show that $a a^\ast \ne {\mathbf 0}_A$.
By the [[Definition:C* Identity|$\text C^\ast$ identity]], we have:
:... | Non-Zero C*-Algebra contains Non-Zero Hermitian Element/Proof 1 | https://proofwiki.org/wiki/Non-Zero_C*-Algebra_contains_Non-Zero_Hermitian_Element | https://proofwiki.org/wiki/Non-Zero_C*-Algebra_contains_Non-Zero_Hermitian_Element/Proof_1 | [
"Non-Zero C*-Algebra contains Non-Zero Hermitian Element",
"C*-Algebras",
"Hermitian Elements of *-Algebras",
"Non-Zero C*-Algebra contains Non-Zero Hermitian Element"
] | [
"Definition:C*-Algebra",
"Definition:Zero Vector",
"Definition:Hermitian Element of *-Algebra"
] | [
"Product of Element in *-Star Algebra with its Star is Hermitian",
"Definition:Hermitian Element of *-Algebra",
"Definition:C* Identity"
] |
proofwiki-22053 | Non-Zero C*-Algebra contains Non-Zero Hermitian Element | Let $\tuple {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$ algebra such that:
:$A \ne \set { {\mathbf 0}_A}$
where ${\mathbf 0}_A$ is the zero vector of $A$.
Then there exists a Hermitian element $x \in A \setminus \set { {\mathbf 0}_A}$. | {{AimForCont}} that ${\mathbf 0}_A$ is the only Hermitian element of $A$.
Let $a \in A$.
From Element of *-Algebra Uniquely Decomposes into Hermitian Elements, there exists Hermitian elements $b, c$ such that:
:$a = b + ic$
Since ${\mathbf 0}_A$ is the only Hermitian element of $A$, we have $b = {\mathbf 0}_A$ and $... | Let $\tuple {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$ algebra]] such that:
:$A \ne \set { {\mathbf 0}_A}$
where ${\mathbf 0}_A$ is the [[Definition:Zero Vector|zero vector]] of $A$.
Then there exists a [[Definition:Hermitian Element of *-Algebra|Hermitian element]] $x \in A \setminu... | {{AimForCont}} that ${\mathbf 0}_A$ is the only [[Definition:Hermitian Element of *-Algebra|Hermitian element]] of $A$.
Let $a \in A$.
From [[Element of *-Algebra Uniquely Decomposes into Hermitian Elements]], there exists [[Definition:Hermitian Element of *-Algebra|Hermitian elements]] $b, c$ such that:
:$a = b + ... | Non-Zero C*-Algebra contains Non-Zero Hermitian Element/Proof 2 | https://proofwiki.org/wiki/Non-Zero_C*-Algebra_contains_Non-Zero_Hermitian_Element | https://proofwiki.org/wiki/Non-Zero_C*-Algebra_contains_Non-Zero_Hermitian_Element/Proof_2 | [
"Non-Zero C*-Algebra contains Non-Zero Hermitian Element",
"C*-Algebras",
"Hermitian Elements of *-Algebras",
"Non-Zero C*-Algebra contains Non-Zero Hermitian Element"
] | [
"Definition:C*-Algebra",
"Definition:Zero Vector",
"Definition:Hermitian Element of *-Algebra"
] | [
"Definition:Hermitian Element of *-Algebra",
"Element of *-Algebra Uniquely Decomposes into Hermitian Elements",
"Definition:Hermitian Element of *-Algebra",
"Definition:Hermitian Element of *-Algebra",
"Definition:Hermitian Element of *-Algebra"
] |
proofwiki-22054 | Spectrum of Commutative C*-Algebra is Non-Empty | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a commutative $\text C^\ast$-algebra such that:
:$A \ne \set { {\mathbf 0}_A}$
where ${\mathbf 0}_A$ is the zero vector of $A$.
Let $\Phi_A$ be the spectrum of $A$.
Then $\Phi_A \ne \O$. | Let $\map {\sigma_A} a$ be the spectrum of $a$ in $A$.
Let $r_A$ denotes spectral radius in $A$.
From Non-Zero C*-Algebra contains Non-Zero Hermitian Element:
:there exists a Hermitian element $x \in A \setminus \set { {\mathbf 0}_A}$.
Hence from {{NormAxiomVector|1}}, we have $\norm a \ne 0$.
From Spectral Radius of ... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Commutative Algebra (Abstract Algebra)|commutative]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]] such that:
:$A \ne \set { {\mathbf 0}_A}$
where ${\mathbf 0}_A$ is the [[Definition:Zero Vector|zero vector]] of $A$.
Let $\Phi_A$ be the [[Definition:Sp... | Let $\map {\sigma_A} a$ be the [[Definition:Spectrum (Spectral Theory)|spectrum]] of $a$ in $A$.
Let $r_A$ denotes [[Definition:Spectral Radius|spectral radius]] in $A$.
From [[Non-Zero C*-Algebra contains Non-Zero Hermitian Element]]:
:there exists a [[Definition:Hermitian Element of *-Algebra|Hermitian element]] $x... | Spectrum of Commutative C*-Algebra is Non-Empty | https://proofwiki.org/wiki/Spectrum_of_Commutative_C*-Algebra_is_Non-Empty | https://proofwiki.org/wiki/Spectrum_of_Commutative_C*-Algebra_is_Non-Empty | [
"C*-Algebras"
] | [
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:C*-Algebra",
"Definition:Zero Vector",
"Definition:Spectrum of Banach Algebra"
] | [
"Definition:Spectrum (Spectral Theory)",
"Definition:Spectral Radius",
"Non-Zero C*-Algebra contains Non-Zero Hermitian Element",
"Definition:Hermitian Element of *-Algebra",
"Spectral Radius of Normal Element of C*-Algebra Equal to Norm",
"Spectrum of Element of Unital Commutative Banach Algebra",
"Cat... |
proofwiki-22055 | Gelfand-Naimark Theorem/Commutative Case | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a commutative $\text C^\ast$-algebra.
Let $\Phi_A$ be the spectrum of $A$.
Let $\struct {\map {\CC_0} {\Phi_A}, \overline \cdot, \norm {\, \cdot \,} }$ be the $\text C^\ast$-algebra of continuous complex-valued functions vanishing at infinity.
Define $G : A \to \map {\CC... | First, from Spectrum of Commutative C*-Algebra is Non-Empty, we have $\Phi_A \ne \O$ and hence the Gelfand transform is indeed defined.
From Spectrum of Banach Algebra is Weak-* Locally Compact Hausdorff Space, $\Phi_A$ is a locally compact Hausdorff space, so we can understand $\map {\CC_0} {\Phi_A}$.
From the Gelfand... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Commutative Algebra (Abstract Algebra)|commutative]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\Phi_A$ be the [[Definition:Spectrum of Banach Algebra|spectrum]] of $A$.
Let $\struct {\map {\CC_0} {\Phi_A}, \overline \cdot, \norm {\, \cdot \,... | First, from [[Spectrum of Commutative C*-Algebra is Non-Empty]], we have $\Phi_A \ne \O$ and hence the [[Definition:Gelfand Transform|Gelfand transform]] is indeed defined.
From [[Spectrum of Banach Algebra is Weak-* Locally Compact Hausdorff Space]], $\Phi_A$ is a [[Definition:Locally Compact Hausdorff Space|locally ... | Gelfand-Naimark Theorem/Commutative Case | https://proofwiki.org/wiki/Gelfand-Naimark_Theorem/Commutative_Case | https://proofwiki.org/wiki/Gelfand-Naimark_Theorem/Commutative_Case | [
"Gelfand-Naimark Theorem"
] | [
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:C*-Algebra",
"Definition:Spectrum of Banach Algebra",
"Definition:C*-Algebra",
"Definition:Continuous Mapping",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Gelfand Transform",
"Definition:Isometric Isomorphism... | [
"Spectrum of Commutative C*-Algebra is Non-Empty",
"Definition:Gelfand Transform",
"Spectrum of Banach Algebra is Weak-* Locally Compact Hausdorff Space",
"Definition:Locally Compact Hausdorff Space",
"Gelfand Representation Theorem",
"Definition:Algebra Homomorphism",
"Definition:Commutative Algebra (A... |
proofwiki-22056 | Gelfand-Naimark Theorem/Commutative Case/Unital | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a commutative unital $\text C^\ast$-algebra.
Let $\Phi_A$ be the spectrum of $A$.
Let $\struct {\map \CC {\Phi_A}, \overline \cdot, \norm {\, \cdot \,} }$ be the $\text C^\ast$-algebra of continuous complex-valued functions vanishing at infinity.
Define $G : A \to \map \... | From Complex-Valued Function on Compact Hausdorff Space Vanishes at Infinity, we have:
:$\map {\CC_0} K = \map \CC K$
for every compact Hausdorff space $K$.
Hence from Space of Continuous Functions Vanishing at Infinity is C*-Algebra, $\struct {\map \CC K, \overline \cdot, \norm {\, \cdot \,} }$ is indeed a $\text C^\a... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Commutative Algebra (Abstract Algebra)|commutative]] [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\Phi_A$ be the [[Definition:Spectrum of Banach Algebra|spectrum]] of $A$.
Let $\struct {\map \CC {\Phi... | From [[Complex-Valued Function on Compact Hausdorff Space Vanishes at Infinity]], we have:
:$\map {\CC_0} K = \map \CC K$
for every [[Definition:Compact Topological Space|compact]] [[Definition:Hausdorff Space|Hausdorff space]] $K$.
Hence from [[Space of Continuous Functions Vanishing at Infinity is C*-Algebra]], $\st... | Gelfand-Naimark Theorem/Commutative Case/Unital | https://proofwiki.org/wiki/Gelfand-Naimark_Theorem/Commutative_Case/Unital | https://proofwiki.org/wiki/Gelfand-Naimark_Theorem/Commutative_Case/Unital | [
"Gelfand-Naimark Theorem"
] | [
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Spectrum of Banach Algebra",
"Definition:C*-Algebra",
"Definition:Continuous Mapping",
"Definition:Space of Continuous Functions on Compact Hausdorff Space",
"Definition:Gelfan... | [
"Complex-Valued Function on Compact Hausdorff Space Vanishes at Infinity",
"Definition:Compact Topological Space",
"Definition:T2 Space",
"Space of Continuous Functions Vanishing at Infinity is C*-Algebra",
"Definition:C*-Algebra",
"Definition:Compact Topological Space",
"Definition:T2 Space",
"Defini... |
proofwiki-22057 | Proof using Axiom of Choice is Nonconstructive | A proof which depends upon the {{Axiom-link|Choice}} is a nonconstructive proof. | The {{Axiom-link|Choice}} allows that an infinite number of selections be made without the need to specify the choice function.
Hence such a proof is nonconstructive.
{{qed}} | A [[Definition:Proof|proof]] which [[Definition:Depend|depends]] upon the {{Axiom-link|Choice}} is a [[Definition:Nonconstructive Proof|nonconstructive proof]]. | The {{Axiom-link|Choice}} allows that an [[Definition:Infinite Set|infinite number]] of selections be made without the need to specify the [[Definition:Choice Function|choice function]].
Hence such a proof is [[Definition:Nonconstructive Proof|nonconstructive]].
{{qed}} | Proof using Axiom of Choice is Nonconstructive | https://proofwiki.org/wiki/Proof_using_Axiom_of_Choice_is_Nonconstructive | https://proofwiki.org/wiki/Proof_using_Axiom_of_Choice_is_Nonconstructive | [
"Axiom of Choice",
"Nonconstructive Proofs"
] | [
"Definition:Proof",
"Definition:Depend",
"Definition:Nonconstructive Proof"
] | [
"Definition:Infinite Set",
"Definition:Choice Function",
"Definition:Nonconstructive Proof"
] |
proofwiki-22058 | Star of Product of Elements in *-Algebra | Let $\tuple {A, \ast}$ be a $\ast$-algebra.
Let $n \ge 2$.
Let $x_1, \ldots, x_n \in A$.
Then:
:$\ds \paren {\prod_{j \mathop = 1}^n x_j}^\ast = \prod_{j \mathop = 1}^n x_{n - j + 1}^\ast$ | We proceed by induction on $n$.
For all $n \in \N$, let $\map P n$ be the proposition:
:$\ds \paren {\prod_{j \mathop = 1}^n x_j}^\ast = \prod_{j \mathop = 1}^n x_{n - j + 1}^\ast$ for all $x_1, \ldots, x_n \in A$. | Let $\tuple {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]].
Let $n \ge 2$.
Let $x_1, \ldots, x_n \in A$.
Then:
:$\ds \paren {\prod_{j \mathop = 1}^n x_j}^\ast = \prod_{j \mathop = 1}^n x_{n - j + 1}^\ast$ | We proceed by [[Principle of Mathematical Induction|induction]] on $n$.
For all $n \in \N$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \paren {\prod_{j \mathop = 1}^n x_j}^\ast = \prod_{j \mathop = 1}^n x_{n - j + 1}^\ast$ for all $x_1, \ldots, x_n \in A$. | Star of Product of Elements in *-Algebra | https://proofwiki.org/wiki/Star_of_Product_of_Elements_in_*-Algebra | https://proofwiki.org/wiki/Star_of_Product_of_Elements_in_*-Algebra | [
"*-Algebras"
] | [
"Definition:*-Algebra"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition"
] |
proofwiki-22059 | Characterization of Localic Mapping Induced by Continuous Mapping | Let $T_1 = \struct{S_1, \tau_1}, T_2 = \struct{S_2, \tau_2}$ be topological spaces.
Let $f: T_1 \to \T_2$ be a continuous mapping.
Let $\map \Omega {T_1}, \map \Omega {T_2}$ be the locales of $T_1$ and $T_2$ respectively.
Let $F : \map \Omega {T_1} \to \map \Omega {T_2}$ be the localic mapping induced by $f$.
Then:
:$\... | Let $G : \map \Omega {T_2} \to \map \Omega {T_1}$ be the frame homomorphism of $f$.
By definition of localic mapping induced by $f$, $F$ is the upper adjoint to $G$.
Hence:
:$\tuple{F, G}$ is a Galois connection
We have:
{{begin-eqn}}
{{eqn | q = \forall U \in \tau_1, V \in \tau_2
| l = V
| o = \subseteq
... | Let $T_1 = \struct{S_1, \tau_1}, T_2 = \struct{S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $f: T_1 \to \T_2$ be a [[Definition:Continuous Mapping (Topological Spaces)|continuous mapping]].
Let $\map \Omega {T_1}, \map \Omega {T_2}$ be the [[Definition:Locale of Topological Space|locales... | Let $G : \map \Omega {T_2} \to \map \Omega {T_1}$ be the [[Definition:Frame Homomorphism of Continuous Mapping|frame homomorphism]] of $f$.
By definition of [[Definition:Localic Mapping Induced by Continuous Mapping|localic mapping induced by $f$]], $F$ is the [[Definition:Upper Adjoint|upper adjoint]] to $G$.
Henc... | Characterization of Localic Mapping Induced by Continuous Mapping | https://proofwiki.org/wiki/Characterization_of_Localic_Mapping_Induced_by_Continuous_Mapping | https://proofwiki.org/wiki/Characterization_of_Localic_Mapping_Induced_by_Continuous_Mapping | [
"Localic Mappings"
] | [
"Definition:Topological Space",
"Definition:Continuous Mapping (Topology)",
"Definition:Locale of Topological Space",
"Definition:Localic Mapping Induced by Continuous Mapping",
"Definition:Relative Complement",
"Definition:Closure (Topology)",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definit... | [
"Definition:Frame Homomorphism of Continuous Mapping",
"Definition:Localic Mapping Induced by Continuous Mapping",
"Definition:Galois Connection/Upper Adjoint",
"Definition:Galois Connection",
"Preimage is Subset Iff Subset of Complement of Image of Complement",
"Set is Subset of Itself",
"Relative Comp... |
proofwiki-22060 | Image is Subset Iff Subset of Preimage | Let $S, T$ be sets.
Let $f: S \to T$ be a mapping.
Then:
:$\forall A \subseteq S, B \subseteq T : f \sqbrk A \subseteq B$ {{iff}} $A \subseteq f^{-1} \sqbrk B$ | === Necessary Condition ===
We have:
{{begin-eqn}}
{{eqn | q = \forall A \subseteq S, B \subseteq T
| l = f \sqbrk A
| o = \subseteq
| r = B
}}
{{eqn | ll = \leadsto
| l = f^{-1} \sqbrk {f \sqbrk A}
| o = \subseteq
| r = f^{-1} \sqbrk B
| c = Preimage of Subset is Subset of Pr... | Let $S, T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Then:
:$\forall A \subseteq S, B \subseteq T : f \sqbrk A \subseteq B$ {{iff}} $A \subseteq f^{-1} \sqbrk B$ | === Necessary Condition ===
We have:
{{begin-eqn}}
{{eqn | q = \forall A \subseteq S, B \subseteq T
| l = f \sqbrk A
| o = \subseteq
| r = B
}}
{{eqn | ll = \leadsto
| l = f^{-1} \sqbrk {f \sqbrk A}
| o = \subseteq
| r = f^{-1} \sqbrk B
| c = [[Preimage of Subset is Subset of... | Image is Subset Iff Subset of Preimage | https://proofwiki.org/wiki/Image_is_Subset_Iff_Subset_of_Preimage | https://proofwiki.org/wiki/Image_is_Subset_Iff_Subset_of_Preimage | [
"Subsets",
"Images",
"Preimages"
] | [
"Definition:Set",
"Definition:Mapping"
] | [
"Preimage of Subset is Subset of Preimage",
"Subset of Domain is Subset of Preimage of Image"
] |
proofwiki-22061 | Inverse Image Mapping is Upper Adjoint to Direct Image Mapping | Let $S, T$ be sets.
Let $f: S \to T$ be a mapping.
Let $f^\to : \powerset S \to \powerset T$ denote the direct image mapping.
Let $f^\gets : \powerset T \to \powerset S$ denote the inverse image mapping.
Then:
:$f^\gets : \powerset T \to \powerset S$ is the upper adjoint to $f^\to : \powerset S \to \powerset T$
That is... | From Image is Subset Iff Subset of Preimage:
:$\forall A \subseteq S, B \subseteq T : f \sqbrk A \subseteq B$ {{iff}} $A \subseteq f^{-1} \sqbrk B$
By definition of direct image mapping:
:$\forall A \subseteq S : \map {f^\to} A = f \sqbrk A$
By definition of inverse image mapping:
:$\forall B \subseteq T : \map {f^\g... | Let $S, T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $f^\to : \powerset S \to \powerset T$ denote the [[Definition:Direct Image Mapping of Mapping|direct image mapping]].
Let $f^\gets : \powerset T \to \powerset S$ denote the [[Definition:Inverse Image Mapping/Mapping|inv... | From [[Image is Subset Iff Subset of Preimage]]:
:$\forall A \subseteq S, B \subseteq T : f \sqbrk A \subseteq B$ {{iff}} $A \subseteq f^{-1} \sqbrk B$
By definition of [[Definition:Direct Image Mapping of Mapping|direct image mapping]]:
:$\forall A \subseteq S : \map {f^\to} A = f \sqbrk A$
By definition of [[Def... | Inverse Image Mapping is Upper Adjoint to Direct Image Mapping | https://proofwiki.org/wiki/Inverse_Image_Mapping_is_Upper_Adjoint_to_Direct_Image_Mapping | https://proofwiki.org/wiki/Inverse_Image_Mapping_is_Upper_Adjoint_to_Direct_Image_Mapping | [
"Direct Image Mappings",
"Inverse Image Mappings",
"Galois Connections"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Direct Image Mapping/Mapping",
"Definition:Inverse Image Mapping/Mapping",
"Definition:Galois Connection/Upper Adjoint"
] | [
"Image is Subset Iff Subset of Preimage",
"Definition:Direct Image Mapping/Mapping",
"Definition:Inverse Image Mapping/Mapping",
"Definition:Galois Connection",
"Definition:Galois Connection",
"Definition:Galois Connection/Upper Adjoint",
"Category:Direct Image Mappings",
"Category:Inverse Image Mappi... |
proofwiki-22062 | Preimage is Subset Iff Subset of Complement of Image of Complement | Let $S, T$ be sets.
Let $f: S \to T$ be a mapping.
Then:
:$\forall A \subseteq S, B \subseteq T : f^{-1} \sqbrk B \subseteq A$ {{iff}} $B \subseteq T \setminus f \sqbrk {S \setminus A}$ | We have:
{{begin-eqn}}
{{eqn | q = \forall A \subseteq S, B \subseteq T
| l = f^{-1} \sqbrk B
| o = \subseteq
| r = A
}}
{{eqn | ll= \leadstoandfrom
| l = S \setminus A
| o = \subseteq
| r = S \setminus {f^{-1} \sqbrk B}
| c = Relative Complement inverts Subsets
}}
{{eqn | r = ... | Let $S, T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Then:
:$\forall A \subseteq S, B \subseteq T : f^{-1} \sqbrk B \subseteq A$ {{iff}} $B \subseteq T \setminus f \sqbrk {S \setminus A}$ | We have:
{{begin-eqn}}
{{eqn | q = \forall A \subseteq S, B \subseteq T
| l = f^{-1} \sqbrk B
| o = \subseteq
| r = A
}}
{{eqn | ll= \leadstoandfrom
| l = S \setminus A
| o = \subseteq
| r = S \setminus {f^{-1} \sqbrk B}
| c = [[Relative Complement inverts Subsets]]
}}
{{eqn | ... | Preimage is Subset Iff Subset of Complement of Image of Complement | https://proofwiki.org/wiki/Preimage_is_Subset_Iff_Subset_of_Complement_of_Image_of_Complement | https://proofwiki.org/wiki/Preimage_is_Subset_Iff_Subset_of_Complement_of_Image_of_Complement | [
"Relative Complement",
"Images",
"Preimages"
] | [
"Definition:Set",
"Definition:Mapping"
] | [
"Relative Complement inverts Subsets",
"Complement of Preimage equals Preimage of Complement",
"Image is Subset Iff Subset of Preimage",
"Relative Complement inverts Subsets",
"Relative Complement of Relative Complement",
"Category:Relative Complement",
"Category:Images",
"Category:Preimages"
] |
proofwiki-22063 | Inverse Image Mapping is Lower Adjoint to Composite Involving Direct Image Mapping | Let $S, T$ be sets.
Let $f: S \to T$ be a mapping.
Let $f^\to : \powerset S \to \powerset T$ denote the direct image mapping.
Let $f^\gets : \powerset T \to \powerset S$ denote the inverse image mapping.
Let $\complement_S: \powerset S \to \powerset S$ denote the relative complement mapping on the power set of $S$.
Let... | We have:
{{begin-eqn}}
{{eqn | q = \forall A \subseteq S, B \subseteq T
| l = \map {f^\gets} B
| o = \subseteq
| r = A
}}
{{eqn | ll = \leadstoandfrom
| l = f^{-1} \sqbrk B
| o = \subseteq
| r = A
| c = {{Defof|Inverse Image Mapping}}
}}
{{eqn | ll = \leadstoandfrom
| l =... | Let $S, T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $f^\to : \powerset S \to \powerset T$ denote the [[Definition:Direct Image Mapping of Mapping|direct image mapping]].
Let $f^\gets : \powerset T \to \powerset S$ denote the [[Definition:Inverse Image Mapping/Mapping|inv... | We have:
{{begin-eqn}}
{{eqn | q = \forall A \subseteq S, B \subseteq T
| l = \map {f^\gets} B
| o = \subseteq
| r = A
}}
{{eqn | ll = \leadstoandfrom
| l = f^{-1} \sqbrk B
| o = \subseteq
| r = A
| c = {{Defof|Inverse Image Mapping}}
}}
{{eqn | ll = \leadstoandfrom
| l =... | Inverse Image Mapping is Lower Adjoint to Composite Involving Direct Image Mapping | https://proofwiki.org/wiki/Inverse_Image_Mapping_is_Lower_Adjoint_to_Composite_Involving_Direct_Image_Mapping | https://proofwiki.org/wiki/Inverse_Image_Mapping_is_Lower_Adjoint_to_Composite_Involving_Direct_Image_Mapping | [
"Images",
"Preimages"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Direct Image Mapping/Mapping",
"Definition:Inverse Image Mapping/Mapping",
"Definition:Relative Complement",
"Definition:Power Set",
"Definition:Relative Complement",
"Definition:Power Set",
"Definition:Galois Connection/Lower Adjoint"
] | [
"Preimage is Subset Iff Subset of Complement of Image of Complement",
"Definition:Galois Connection",
"Definition:Galois Connection",
"Definition:Galois Connection/Lower Adjoint",
"Category:Images",
"Category:Preimages"
] |
proofwiki-22064 | Norm satisfying Parallelogram Law induced by Inner Product/Complex Case | Let $V$ be a vector space over $\R$.
Let $\norm \cdot : V \to \R$ be a norm on $V$ such that:
:$\norm {x + y}^2 + \norm {x - y}^2 = 2 \paren {\norm x^2 + \norm y^2}$
for each $x, y \in V$.
Then the function $\innerprod \cdot \cdot : V \times V \to \C$ defined by:
:$\ds \innerprod x y = \frac 1 4 \sum_{k \mathop = 0}^... | We write out:
:$\innerprod x y = \dfrac 1 4 \paren {\norm {x + y}^2 + i \norm {x + i y}^2 - \norm {x - y}^2 - i \norm {x - i y}^2}$
We show that $\innerprod \cdot \cdot$ is an inner product. | Let $V$ be a [[Definition:Vector Space|vector space]] over $\R$.
Let $\norm \cdot : V \to \R$ be a [[Definition:Norm on Vector Space|norm]] on $V$ such that:
:$\norm {x + y}^2 + \norm {x - y}^2 = 2 \paren {\norm x^2 + \norm y^2}$
for each $x, y \in V$.
Then the [[Definition:Function|function]] $\innerprod \cdot \... | We write out:
:$\innerprod x y = \dfrac 1 4 \paren {\norm {x + y}^2 + i \norm {x + i y}^2 - \norm {x - y}^2 - i \norm {x - i y}^2}$
We show that $\innerprod \cdot \cdot$ is an [[Definition:Inner Product|inner product]]. | Norm satisfying Parallelogram Law induced by Inner Product/Complex Case | https://proofwiki.org/wiki/Norm_satisfying_Parallelogram_Law_induced_by_Inner_Product/Complex_Case | https://proofwiki.org/wiki/Norm_satisfying_Parallelogram_Law_induced_by_Inner_Product/Complex_Case | [
"Norm satisfying Parallelogram Law induced by Inner Product"
] | [
"Definition:Vector Space",
"Definition:Norm/Vector Space",
"Definition:Function",
"Definition:Inner Product",
"Definition:Inner Product Norm"
] | [
"Definition:Inner Product",
"Definition:Inner Product"
] |
proofwiki-22065 | Interior of Set of Real Numbers in Complex Numbers is Empty | Let $\tuple {\C, d}$ be the complex Euclidean space.
Let $\R$ be the subspace of real numbers.
Then the interior of $\R$ in $\C$ is the empty set $\O$. | {{AimForCont}} that:
:$\R^\circ \ne \O$
Let $x \in \R^\circ$.
From the definition of an open subset of $\C$, there exists $\epsilon \in \R_{>0}$ such that:
:$\set {z \in \C : \cmod {z - x} < \epsilon} \subseteq \R^\circ$
Consider:
:$z = x + \dfrac \epsilon 2 i \in \C \setminus \R$
Then, we have:
:$\cmod {z - x} = \cmo... | Let $\tuple {\C, d}$ be the [[Definition:Euclidean Space/Complex|complex Euclidean space]].
Let $\R$ be the [[Definition:Topological Subspace|subspace]] of [[Definition:Real Number|real numbers]].
Then the [[Definition:Interior (Topology)|interior]] of $\R$ in $\C$ is the [[Definition:Empty Set|empty set]] $\O$. | {{AimForCont}} that:
:$\R^\circ \ne \O$
Let $x \in \R^\circ$.
From the definition of an [[Definition:Open Set (Complex Analysis)|open subset of $\C$]], there exists $\epsilon \in \R_{>0}$ such that:
:$\set {z \in \C : \cmod {z - x} < \epsilon} \subseteq \R^\circ$
Consider:
:$z = x + \dfrac \epsilon 2 i \in \C \setm... | Interior of Set of Real Numbers in Complex Numbers is Empty | https://proofwiki.org/wiki/Interior_of_Set_of_Real_Numbers_in_Complex_Numbers_is_Empty | https://proofwiki.org/wiki/Interior_of_Set_of_Real_Numbers_in_Complex_Numbers_is_Empty | [
"Interior of Set of Real Numbers in Complex Numbers is Empty",
"Examples of Set Interiors",
"Complex Numbers",
"Real Numbers"
] | [
"Definition:Euclidean Space/Complex",
"Definition:Topological Subspace",
"Definition:Real Number",
"Definition:Interior (Topology)",
"Definition:Empty Set"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Interior (Topology)",
"Category:Interior of Set of Real Numbers in Complex Numbers is Empty",
"Category:Examples of Set Interiors",
"Category:Complex Numbers",
"Category:Real Numbers"
] |
proofwiki-22066 | Interior of Set of Real Numbers in Complex Numbers is Empty/Corollary | Let $\tuple {\C, d}$ be the complex Euclidean space.
Consider $S \subseteq \R$ as a topological subspace of $\tuple {\C, d}$.
Then the interior of $S$ in $\C$ is the empty set $\O$. | From Interior of Subset we have:
:$S^\circ \subseteq \R^\circ$
From Interior of Set of Real Numbers in Complex Numbers is Empty, we have:
:$\R^\circ = \O$
Hence $S^\circ \subseteq \O$ and so:
:$S^\circ = \O$
from Subset of Empty Set.
{{qed}}
Category:Interior of Set of Real Numbers in Complex Numbers is Empty
d3kct1pun... | Let $\tuple {\C, d}$ be the [[Definition:Euclidean Space/Complex|complex Euclidean space]].
Consider $S \subseteq \R$ as a [[Definition:Topological Subspace|topological subspace]] of $\tuple {\C, d}$.
Then the [[Definition:Interior (Topology)|interior]] of $S$ in $\C$ is the [[Definition:Empty Set|empty set]] $\O$. | From [[Interior of Subset]] we have:
:$S^\circ \subseteq \R^\circ$
From [[Interior of Set of Real Numbers in Complex Numbers is Empty]], we have:
:$\R^\circ = \O$
Hence $S^\circ \subseteq \O$ and so:
:$S^\circ = \O$
from [[Subset of Empty Set]].
{{qed}}
[[Category:Interior of Set of Real Numbers in Complex Numbers i... | Interior of Set of Real Numbers in Complex Numbers is Empty/Corollary | https://proofwiki.org/wiki/Interior_of_Set_of_Real_Numbers_in_Complex_Numbers_is_Empty/Corollary | https://proofwiki.org/wiki/Interior_of_Set_of_Real_Numbers_in_Complex_Numbers_is_Empty/Corollary | [
"Interior of Set of Real Numbers in Complex Numbers is Empty"
] | [
"Definition:Euclidean Space/Complex",
"Definition:Topological Subspace",
"Definition:Interior (Topology)",
"Definition:Empty Set"
] | [
"Interior of Subset",
"Interior of Set of Real Numbers in Complex Numbers is Empty",
"Subset of Empty Set",
"Category:Interior of Set of Real Numbers in Complex Numbers is Empty"
] |
proofwiki-22067 | Localic Functor is Covariant | Let $\mathbf{Top}$ denote the category of topological spaces.
Let $\mathbf{Loc_*}$ denote the category of locales with localic mappings.
Then:
:the localic functor $\mathbf {Lc} : \mathbf{Top} \to \mathbf{Loc_*}$ is a covariant functor | === Object Functor is Well-Defined ===
Let $T$ be a topological space.
By definition of localic object functor:
:$\map {\mathbf {Lc}} T$ is the locale of $T$
From Locale of Topological Space is Locale:
:$\map {\mathbf {Lc}} T$ is a locale
By definition of category of locales with localic mappings:
:$\map {\mathbf {Lc}}... | Let $\mathbf{Top}$ denote the [[Definition:Category of Topological Spaces|category of topological spaces]].
Let $\mathbf{Loc_*}$ denote the [[Definition:Category of Locales with Localic Mappings|category of locales with localic mappings]].
Then:
:the [[Definition:Localic Functor|localic functor]] $\mathbf {Lc} : \ma... | === Object Functor is Well-Defined ===
Let $T$ be a [[Definition:Topological Space|topological space]].
By definition of [[Definition:Localic Functor|localic object functor]]:
:$\map {\mathbf {Lc}} T$ is the [[Definition:Locale of Topological Space|locale]] of $T$
From [[Locale of Topological Space is Locale]]:
:$... | Localic Functor is Covariant | https://proofwiki.org/wiki/Localic_Functor_is_Covariant | https://proofwiki.org/wiki/Localic_Functor_is_Covariant | [
"Functors"
] | [
"Definition:Category of Topological Spaces",
"Definition:Category of Locales with Localic Mappings",
"Definition:Localic Functor",
"Definition:Functor/Covariant"
] | [
"Definition:Topological Space",
"Definition:Localic Functor",
"Definition:Locale of Topological Space",
"Locale of Topological Space is Locale",
"Definition:Locale (Lattice Theory)",
"Definition:Category of Locales with Localic Mappings",
"Definition:Object",
"Definition:Object Functor",
"Definition... |
proofwiki-22068 | Power in Prime Ideal | Let $A$ be a commutative ring with unity.
Let $\mathfrak p \subseteq A$ be a prime ideal.
Let $n > 0$ be a natural number.
Then for all elements $a \in A$, if $a^n \in \mathfrak p$ then $a \in \mathfrak p$. | Proof by induction: | Let $A$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $\mathfrak p \subseteq A$ be a [[Definition:Prime Ideal of Ring|prime ideal]].
Let $n > 0$ be a [[Definition:Natural Number|natural number]].
Then for all elements $a \in A$, if $a^n \in \mathfrak p$ then $a \in \mathfrak p$. | Proof by [[Principle of Mathematical Induction|induction]]: | Power in Prime Ideal | https://proofwiki.org/wiki/Power_in_Prime_Ideal | https://proofwiki.org/wiki/Power_in_Prime_Ideal | [
"Ideal Theory",
"Prime Ideals of Rings"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Prime Ideal of Ring",
"Definition:Natural Numbers"
] | [
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-22069 | Poisson Distribution Approximated by Normal Distribution | Let $X$ be a discrete random variable which has the Poisson distribution $\Poisson \lambda$.
Then for large $\lambda$:
:$\Poisson \lambda \approx \Gaussian \lambda \lambda$
where $\Gaussian \lambda \lambda$ denotes the normal distribution. | {{MissingLinks}}
{{tidy}}
Let $n$ be a sufficiently large number.
Let $X_1,X_2,\dots X_n$ be independent and identically distributed random variables such that they have a $\Poisson {\frac{\lambda}{n}}$ distribution.
It follows from Expectation of Poisson Distribution and Variance of Poisson Distribution that $\mu = \e... | Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]] which has the [[Definition:Poisson Distribution|Poisson distribution $\Poisson \lambda$]].
Then for large $\lambda$:
:$\Poisson \lambda \approx \Gaussian \lambda \lambda$
where $\Gaussian \lambda \lambda$ denotes the [[Definition:Normal Dis... | {{MissingLinks}}
{{tidy}}
Let $n$ be a sufficiently large number.
Let $X_1,X_2,\dots X_n$ be [[Definition:Independent and Identically Distributed|independent and identically distributed]] [[Definition:Random Variable|random variables]] such that they have a [[Definition:Poisson Distribution|$\Poisson {\frac{\lambda}{... | Poisson Distribution Approximated by Normal Distribution | https://proofwiki.org/wiki/Poisson_Distribution_Approximated_by_Normal_Distribution | https://proofwiki.org/wiki/Poisson_Distribution_Approximated_by_Normal_Distribution | [
"Poisson Distribution",
"Normal Distribution"
] | [
"Definition:Random Variable/Discrete",
"Definition:Poisson Distribution",
"Definition:Normal Distribution"
] | [
"Definition:Random Sample (Probability Theory)",
"Definition:Random Variable",
"Definition:Poisson Distribution",
"Expectation of Poisson Distribution",
"Variance of Poisson Distribution",
"Sum of Independent Poisson Random Variables is Poisson",
"Central Limit Theorem",
"Definition:Standard Normal Di... |
proofwiki-22070 | Extension of Continuous Complex-Valued Function Vanishing at Infinity to Alexandroff Extension is Continuous/Corollary | Let $X$ be a locally compact Hausdorff space.
Let $f : X \to \C$ be a continuous complex-valued function vanishing at infinity.
Then $f \sqbrk X \cup \set 0$ is compact. | Let $X^\ast = X \cup \set p$ be the Alexandroff extension of $X$.
From Alexandroff Extension is Compact, $X^\ast$ is compact.
Define $f^\ast : X^\ast \to \C$ by taking:
:$\map {f^\ast} x = \begin{cases}\map f x & x \in X \\ 0 & x = p\end{cases}$
for each $x \in X^\ast$.
From Extension of Continuous Complex-Valued Func... | Let $X$ be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]].
Let $f : X \to \C$ be a [[Definition:Continuous Mapping|continuous]] [[Definition:Complex-Valued Function Vanishing at Infinity|complex-valued function vanishing at infinity]].
Then $f \sqbrk X \cup \set 0$ is [[Definition:C... | Let $X^\ast = X \cup \set p$ be the [[Definition:Alexandroff Extension|Alexandroff extension]] of $X$.
From [[Alexandroff Extension is Compact]], $X^\ast$ is [[Definition:Compact Topological Space|compact]].
Define $f^\ast : X^\ast \to \C$ by taking:
:$\map {f^\ast} x = \begin{cases}\map f x & x \in X \\ 0 & x = p\en... | Extension of Continuous Complex-Valued Function Vanishing at Infinity to Alexandroff Extension is Continuous/Corollary | https://proofwiki.org/wiki/Extension_of_Continuous_Complex-Valued_Function_Vanishing_at_Infinity_to_Alexandroff_Extension_is_Continuous/Corollary | https://proofwiki.org/wiki/Extension_of_Continuous_Complex-Valued_Function_Vanishing_at_Infinity_to_Alexandroff_Extension_is_Continuous/Corollary | [
"Extension of Continuous Complex-Valued Function Vanishing at Infinity to Alexandroff Extension is Continuous"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:Continuous Mapping",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Compact Topological Space"
] | [
"Definition:Alexandroff Extension",
"Alexandroff Extension is Compact",
"Definition:Compact Topological Space",
"Extension of Continuous Complex-Valued Function Vanishing at Infinity to Alexandroff Extension is Continuous",
"Definition:Continuous Function",
"Continuous Image of Compact Space is Compact",
... |
proofwiki-22071 | Reciprocal of Continuous Complex-Valued Function Vanishing at Infinity does not Vanish at Infinity | Let $X$ be a locally compact Hausdorff space.
Let $f : X \to \C$ be a continuous function that vanishes at infinity such that:
:$\map f x \ne 0$ for each $x \in X$.
Then $\dfrac 1 f : X \to \C$ does not vanish at infinity. | From Quotient Rule for Continuous Complex Functions:
:$\dfrac 1 f$ is continuous.
{{AimForCont}} $\dfrac 1 f$ vanishes at infinity.
Then from Continuous Complex-Valued Function Vanishing at Infinity is Bounded and Attains Supremum, $\dfrac 1 f$ is bounded.
Since $f$ vanishes at infinity, for each $n \in \N$ there exist... | Let $X$ be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]].
Let $f : X \to \C$ be a [[Definition:Continuous Function|continuous function]] that [[Definition:Complex-Valued Function Vanishing at Infinity|vanishes at infinity]] such that:
:$\map f x \ne 0$ for each $x \in X$.
Then $\d... | From [[Quotient Rule for Continuous Complex Functions]]:
:$\dfrac 1 f$ is [[Definition:Continuous Function|continuous]].
{{AimForCont}} $\dfrac 1 f$ [[Definition:Complex-Valued Function Vanishing at Infinity|vanishes at infinity]].
Then from [[Continuous Complex-Valued Function Vanishing at Infinity is Bounded and At... | Reciprocal of Continuous Complex-Valued Function Vanishing at Infinity does not Vanish at Infinity | https://proofwiki.org/wiki/Reciprocal_of_Continuous_Complex-Valued_Function_Vanishing_at_Infinity_does_not_Vanish_at_Infinity | https://proofwiki.org/wiki/Reciprocal_of_Continuous_Complex-Valued_Function_Vanishing_at_Infinity_does_not_Vanish_at_Infinity | [
"Complex-Valued Functions Vanishing at Infinity"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:Continuous Function",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Complex-Valued Function Vanishing at Infinity"
] | [
"Combination Theorem for Continuous Functions/Complex/Quotient Rule",
"Definition:Continuous Function",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Continuous Complex-Valued Function Vanishing at Infinity is Bounded and Attains Supremum",
"Definition:Bounded Mapping/Complex-Valued",
"Defi... |
proofwiki-22072 | Unitization of Non-Unital Subalgebra of Unital Algebra is Isomorphic to Unital Subalgebra | Let $A$ be a unital algebra over $\C$.
Let $B$ be a subalgebra of $A$ that is not unital.
Let $B_+$ be the unitization of $B$.
Let $B \oplus \C {\mathbf 1}_A$ be the (internal) direct sum of $B$ and $\C {\mathbf 1}_A$.
Define $f : B_+ \to B \oplus \C {\mathbf 1}_A$ by:
:$\map f {b, \lambda} = b + \lambda {\mathbf 1}_A... | Since:
:$B \oplus {\mathbf 1}_A = \set {b + \lambda {\mathbf 1}_A : b \in B, \, \lambda \in \C} = \set {\map f {b, \lambda} : \tuple {b, \lambda} \in B_+}$
we have that $f$ is a surjection.
We show that $f$ is linear.
Let $\tuple {b_1, \lambda_1}, \tuple {b_2, \lambda_2} \in B_+$ and $t \in \C$.
We then have:
{{begin-e... | Let $A$ be a [[Definition:Unital Algebra|unital algebra]] over $\C$.
Let $B$ be a [[Definition:Subalgebra|subalgebra]] of $A$ that is not [[Definition:Unital Subalgebra|unital]].
Let $B_+$ be the [[Definition:Unitization of Algebra over Field|unitization]] of $B$.
Let $B \oplus \C {\mathbf 1}_A$ be the [[Definition:... | Since:
:$B \oplus {\mathbf 1}_A = \set {b + \lambda {\mathbf 1}_A : b \in B, \, \lambda \in \C} = \set {\map f {b, \lambda} : \tuple {b, \lambda} \in B_+}$
we have that $f$ is a [[Definition:Surjection|surjection]].
We show that $f$ is [[Definition:Linear Transformation|linear]].
Let $\tuple {b_1, \lambda_1}, \tuple ... | Unitization of Non-Unital Subalgebra of Unital Algebra is Isomorphic to Unital Subalgebra | https://proofwiki.org/wiki/Unitization_of_Non-Unital_Subalgebra_of_Unital_Algebra_is_Isomorphic_to_Unital_Subalgebra | https://proofwiki.org/wiki/Unitization_of_Non-Unital_Subalgebra_of_Unital_Algebra_is_Isomorphic_to_Unital_Subalgebra | [
"Algebras",
"Unitizations of Algebras over Fields"
] | [
"Definition:Unital Algebra",
"Definition:Subalgebra",
"Definition:Unital Subalgebra",
"Definition:Unitization of Algebra over Field",
"Definition:Internal Direct Sum of Modules",
"Definition:Unital Algebra Isomorphism"
] | [
"Definition:Surjection",
"Definition:Linear Transformation",
"Definition:Linear Transformation",
"Definition:Algebra Homomorphism",
"Linear Transformation is Injective iff Kernel Contains Only Zero",
"Definition:Unital Subalgebra",
"Definition:Algebra Isomorphism",
"Definition:Unital Algebra Isomorphi... |
proofwiki-22073 | Image of Group of Units in Unital Algebra under Unital Algebra Homomorphism | Let $R$ be a ring.
Let $A, B$ be unital $R$-algebras.
Let $\phi : A \to B$ be an unital algebra homomorphism.
Let $\map G A$ and $\map G B$ the groups of units of $A$ and $B$ respectively.
Then:
:$\phi \sqbrk {\map G A} \subseteq \map G B$ | Let ${\mathbf 1}_A$ be the identity element of $A$.
Let $x \in \map G A$.
Then there exists $y \in A$ such that $x y = y x = {\mathbf 1}_A$.
We then have $\map \phi {x y} = \map \phi {y x} = \map \phi { {\mathbf 1}_A} = {\mathbf 1}_B$ since $\phi$ is a unital algebra homomorphism.
Since $\phi$ is an algebra homomorph... | Let $R$ be a [[Definition:Ring|ring]].
Let $A, B$ be [[Definition:Unital Algebra|unital $R$-algebras]].
Let $\phi : A \to B$ be an [[Definition:Unital Algebra Homomorphism|unital algebra homomorphism]].
Let $\map G A$ and $\map G B$ the [[Definition:Group of Units|groups of units]] of $A$ and $B$ respectively.
Th... | Let ${\mathbf 1}_A$ be the [[Definition:Identity Element|identity element]] of $A$.
Let $x \in \map G A$.
Then there exists $y \in A$ such that $x y = y x = {\mathbf 1}_A$.
We then have $\map \phi {x y} = \map \phi {y x} = \map \phi { {\mathbf 1}_A} = {\mathbf 1}_B$ since $\phi$ is a [[Definition:Unital Algebra Ho... | Image of Group of Units in Unital Algebra under Unital Algebra Homomorphism | https://proofwiki.org/wiki/Image_of_Group_of_Units_in_Unital_Algebra_under_Unital_Algebra_Homomorphism | https://proofwiki.org/wiki/Image_of_Group_of_Units_in_Unital_Algebra_under_Unital_Algebra_Homomorphism | [
"Algebras",
"Image of Group of Units in Unital Algebra under Unital Algebra Homomorphism"
] | [
"Definition:Ring",
"Definition:Unital Algebra",
"Definition:Unital Algebra Homomorphism",
"Definition:Group of Units"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Unital Algebra Homomorphism",
"Definition:Algebra Homomorphism",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Category:Algebras",
"Category:Image of Group of Units in Unital Algebra under Unital Algebra Homomorphism"
] |
proofwiki-22074 | Spectrum of Image of Element of Unital Algebra under Unital Algebra Homomorphism | Let $A, B$ be unital algebras over $\C$.
Let $\phi : A \to B$ be a unital algebra homomorphism.
Let $x \in A$.
We then have:
:$\map {\sigma_B} {\map \phi x} \subseteq \map {\sigma_A} x$
where $\sigma_A$ and $\sigma_B$ denote spectra in $A$ and $B$ respectively.
If $\phi$ is a unital algebra isomorphism, we further ha... | We show that:
:$\map {\rho_A} x \subseteq \map {\rho_B} {\map \phi x}$
where $\rho_A$ and $\rho_B$ denote the resolvent sets in $A$ and $B$ respectively.
Let $\lambda \in \map {\rho_A} x$.
Then there exists $y \in A$ such that $\paren {\lambda {\mathbf 1}_A - x} y = y \paren {\lambda {\mathbf 1}_A - x} = {\mathbf 1}_A$... | Let $A, B$ be [[Definition:Unital Algebra|unital algebras]] over $\C$.
Let $\phi : A \to B$ be a [[Definition:Unital Algebra Homomorphism|unital algebra homomorphism]].
Let $x \in A$.
We then have:
:$\map {\sigma_B} {\map \phi x} \subseteq \map {\sigma_A} x$
where $\sigma_A$ and $\sigma_B$ denote [[Definition:Spec... | We show that:
:$\map {\rho_A} x \subseteq \map {\rho_B} {\map \phi x}$
where $\rho_A$ and $\rho_B$ denote the [[Definition:Resolvent Set|resolvent sets]] in $A$ and $B$ respectively.
Let $\lambda \in \map {\rho_A} x$.
Then there exists $y \in A$ such that $\paren {\lambda {\mathbf 1}_A - x} y = y \paren {\lambda {\ma... | Spectrum of Image of Element of Unital Algebra under Unital Algebra Homomorphism | https://proofwiki.org/wiki/Spectrum_of_Image_of_Element_of_Unital_Algebra_under_Unital_Algebra_Homomorphism | https://proofwiki.org/wiki/Spectrum_of_Image_of_Element_of_Unital_Algebra_under_Unital_Algebra_Homomorphism | [
"Spectrum of Image of Element of Unital Algebra under Unital Algebra Homomorphism",
"Spectra (Spectral Theory)"
] | [
"Definition:Unital Algebra",
"Definition:Unital Algebra Homomorphism",
"Definition:Spectrum (Spectral Theory)/Unital Algebra",
"Definition:Unital Algebra Isomorphism"
] | [
"Definition:Resolvent Set",
"Definition:Unital Algebra Homomorphism",
"Definition:Algebra Homomorphism",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Algebra Homomorphism",
"Definition:Unital Algebra Isomorphism",
"Definition:Unital Algebra Isomorphism",
"Inverse of Algebraic Structure... |
proofwiki-22075 | Spectrum of Element of Space of Continuous Functions Vanishing at Infinity | Let $X$ be a locally compact Hausdorff space.
Let $A = \map {\CC_0} X$ be the Banach algebra of continuous complex-valued functions vanishing at infinity.
Let $f \in \map {\CC_0} X$.
Then:
:$\map {\sigma_A} f = f \sqbrk X \cup \set 0$
where $\sigma_A$ denotes the spectrum in $A$. | Let $\map {\CC_b} X$ be the Banach algebra of bounded continuous functions on $X$ valued in $\C$.
Define $\mathbf 1 : X \to \C$ by:
:$\map {\mathbf 1} x = 1$ for each $x \in X$.
Then $\mathbf 1$ is the identity element for $\map {\CC_b} X$.
Let $A_+$ be the unitization of $A$.
Let $A \oplus \C \mathbf 1$ be the intern... | Let $X$ be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]].
Let $A = \map {\CC_0} X$ be the [[Definition:Banach Algebra|Banach algebra]] of [[Definition:Continuous Mapping|continuous]] [[Definition:Complex-Valued Function Vanishing at Infinity|complex-valued functions vanishing at infi... | Let $\map {\CC_b} X$ be the [[Definition:Banach Algebra|Banach algebra]] of [[Definition:Space of Bounded Continuous Functions on Topological Space|bounded continuous functions on $X$ valued in $\C$]].
Define $\mathbf 1 : X \to \C$ by:
:$\map {\mathbf 1} x = 1$ for each $x \in X$.
Then $\mathbf 1$ is the [[Definition... | Spectrum of Element of Space of Continuous Functions Vanishing at Infinity | https://proofwiki.org/wiki/Spectrum_of_Element_of_Space_of_Continuous_Functions_Vanishing_at_Infinity | https://proofwiki.org/wiki/Spectrum_of_Element_of_Space_of_Continuous_Functions_Vanishing_at_Infinity | [
"Banach Algebras"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:Banach Algebra",
"Definition:Continuous Mapping",
"Definition:Complex-Valued Function Vanishing at Infinity",
"Definition:Spectrum (Spectral Theory)/Non-Unital Algebra"
] | [
"Definition:Banach Algebra",
"Definition:Space of Bounded Continuous Functions on Topological Space",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Unitization of Algebra over Field",
"Definition:Internal Direct Sum of Modules",
"Unitization of Non-Unital Subalgebra of Unital Al... |
proofwiki-22076 | Subalgebra Generated by Self-Adjoint Set is Self-Adjoint | Let $\tuple {A, \ast}$ be a $\ast$-algebra over $\C$.
Let $S \subseteq A$ be self-adjoint.
Let $K \sqbrk S$ be the subalgebra generated by $S$.
Then $\C \sqbrk S$ is a $\ast$-subalgebra of $A$. | From Explicit Form for Generated Subalgebra, we have:
:$\C \sqbrk S = \map \span \AA$
where:
:$\AA = \set {x_1^{k_1} x_2^{k_2} \ldots x_n^{k_n} : x_1, \ldots, x_n \in S, \, k_1, \ldots, k_n \ge 1}$
From Star of Product of Elements in *-Algebra we have:
:$\paren {x_1^{k_1} x_2^{k_2} \ldots x_n^{k_n} }^\ast = x_n^{k_n} \... | Let $\tuple {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]] over $\C$.
Let $S \subseteq A$ be [[Definition:Self-Adjoint Subset of *-Algebra|self-adjoint]].
Let $K \sqbrk S$ be the [[Definition:Generated Subalgebra|subalgebra generated by $S$]].
Then $\C \sqbrk S$ is a [[Definition:*-Subalgebra|$\ast$-subal... | From [[Explicit Form for Generated Subalgebra]], we have:
:$\C \sqbrk S = \map \span \AA$
where:
:$\AA = \set {x_1^{k_1} x_2^{k_2} \ldots x_n^{k_n} : x_1, \ldots, x_n \in S, \, k_1, \ldots, k_n \ge 1}$
From [[Star of Product of Elements in *-Algebra]] we have:
:$\paren {x_1^{k_1} x_2^{k_2} \ldots x_n^{k_n} }^\ast = x_... | Subalgebra Generated by Self-Adjoint Set is Self-Adjoint | https://proofwiki.org/wiki/Subalgebra_Generated_by_Self-Adjoint_Set_is_Self-Adjoint | https://proofwiki.org/wiki/Subalgebra_Generated_by_Self-Adjoint_Set_is_Self-Adjoint | [
"*-Algebras"
] | [
"Definition:*-Algebra",
"Definition:Self-Adjoint Subset of *-Algebra",
"Definition:Generator of Algebra",
"Definition:*-Subalgebra"
] | [
"Explicit Form for Generated Subalgebra",
"Star of Product of Elements in *-Algebra",
"Definition:Self-Adjoint Subset of *-Algebra",
"Definition:Self-Adjoint Subset of *-Algebra",
"Linear Span of Self-Adjoint Subset of *-Algebra is Self-Adjoint",
"Definition:*-Subalgebra",
"Category:*-Algebras"
] |
proofwiki-22077 | Linear Span of Self-Adjoint Subset of *-Algebra is Self-Adjoint | Let $\tuple {A, \ast}$ be a $\ast$-algebra over $\C$.
Let $S \subseteq A$ be self-adjoint.
Then $\map \span S$ is self-adjoint. | Let:
:$\ds \sum_{i \mathop = 1}^n \lambda_i x_i \in \map \span S$
where $\lambda_1, \ldots, \lambda_n \in \C$ and $x_1, \ldots, x_n \in S$.
From $(\text C^\ast 2)$ and $(\text C^\ast 3)$ in the definition of an involution, we have:
:$\ds \paren {\sum_{i \mathop = 1}^n \lambda_i x_i}^\ast = \sum_{i \mathop = 1}^n \overl... | Let $\tuple {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]] over $\C$.
Let $S \subseteq A$ be [[Definition:Self-Adjoint Subset of *-Algebra|self-adjoint]].
Then $\map \span S$ is [[Definition:Self-Adjoint Subset of *-Algebra|self-adjoint]]. | Let:
:$\ds \sum_{i \mathop = 1}^n \lambda_i x_i \in \map \span S$
where $\lambda_1, \ldots, \lambda_n \in \C$ and $x_1, \ldots, x_n \in S$.
From $(\text C^\ast 2)$ and $(\text C^\ast 3)$ in the definition of an [[Definition:Involution on Algebra|involution]], we have:
:$\ds \paren {\sum_{i \mathop = 1}^n \lambda_i x_i... | Linear Span of Self-Adjoint Subset of *-Algebra is Self-Adjoint | https://proofwiki.org/wiki/Linear_Span_of_Self-Adjoint_Subset_of_*-Algebra_is_Self-Adjoint | https://proofwiki.org/wiki/Linear_Span_of_Self-Adjoint_Subset_of_*-Algebra_is_Self-Adjoint | [
"*-Algebras"
] | [
"Definition:*-Algebra",
"Definition:Self-Adjoint Subset of *-Algebra",
"Definition:Self-Adjoint Subset of *-Algebra"
] | [
"Definition:Involution on Algebra",
"Definition:Self-Adjoint Subset of *-Algebra",
"Definition:Self-Adjoint Subset of *-Algebra",
"Category:*-Algebras"
] |
proofwiki-22078 | Star of Convergent Sequence in Banach *-Algebra Converges | Let $\struct {A, \ast}$ be a $\ast$-algebra over $\C$.
Let $\sequence {x_n}_{n \in \N}$ be a convergent sequence in $A$ with:
:$x_n \to x$
Then:
:$x_n^\ast \to x^\ast$ | We have:
{{begin-eqn}}
{{eqn | l = \norm {x_n^\ast - x^\ast}
| r = \norm {\paren {x_n - x}^\ast}
| c = $(\text C^\ast 2)$, $(\text C^\ast 4)$ in definition of involution
}}
{{eqn | r = \norm {x_n - x}
| c = {{Defof|Banach *-Algebra}}
}}
{{eqn | o = \to
| r = 0
| c = Sequence in Normed Vector Space Convergent... | Let $\struct {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]] over $\C$.
Let $\sequence {x_n}_{n \in \N}$ be a [[Definition:Convergent Sequence|convergent sequence]] in $A$ with:
:$x_n \to x$
Then:
:$x_n^\ast \to x^\ast$ | We have:
{{begin-eqn}}
{{eqn | l = \norm {x_n^\ast - x^\ast}
| r = \norm {\paren {x_n - x}^\ast}
| c = $(\text C^\ast 2)$, $(\text C^\ast 4)$ in definition of [[Definition:Involution on Algebra|involution]]
}}
{{eqn | r = \norm {x_n - x}
| c = {{Defof|Banach *-Algebra}}
}}
{{eqn | o = \to
| r = 0
| c = [[Seq... | Star of Convergent Sequence in Banach *-Algebra Converges | https://proofwiki.org/wiki/Star_of_Convergent_Sequence_in_Banach_*-Algebra_Converges | https://proofwiki.org/wiki/Star_of_Convergent_Sequence_in_Banach_*-Algebra_Converges | [
"Banach *-Algebras"
] | [
"Definition:*-Algebra",
"Definition:Convergent Sequence"
] | [
"Definition:Involution on Algebra",
"Sequence in Normed Vector Space Convergent to Limit iff Norm of Sequence minus Limit is Null Sequence",
"Category:Banach *-Algebras"
] |
proofwiki-22079 | Closure of Self-Adjoint Subset of Banach *-Algebra is Self-Adjoint | Let $\struct {A, \ast}$ be a $\ast$-algebra over $\C$.
Let $S \subseteq A$ be self-adjoint.
Then the closure $S^-$ of $S$ is self-adjoint. | Let $x \in S^-$.
From Point in Closure of Subset of Metric Space iff Limit of Sequence, there exists a sequence $\sequence {x_n}_{n \mathop \in \N}$ in $S$ with $x_n \to x$.
Since $S$ is self-adjoint, we have $x_n^\ast \in S$.
From Star of Convergent Sequence in Banach *-Algebra Converges, we have $x_n^\ast \to x^\a... | Let $\struct {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]] over $\C$.
Let $S \subseteq A$ be [[Definition:Self-Adjoint Subset of *-Algebra|self-adjoint]].
Then the [[Definition:Topological Closure|closure]] $S^-$ of $S$ is [[Definition:Self-Adjoint Subset of *-Algebra|self-adjoint]]. | Let $x \in S^-$.
From [[Point in Closure of Subset of Metric Space iff Limit of Sequence]], there exists a [[Definition:Sequence|sequence]] $\sequence {x_n}_{n \mathop \in \N}$ in $S$ with $x_n \to x$.
Since $S$ is [[Definition:Self-Adjoint Subset of *-Algebra|self-adjoint]], we have $x_n^\ast \in S$.
From [[Star... | Closure of Self-Adjoint Subset of Banach *-Algebra is Self-Adjoint | https://proofwiki.org/wiki/Closure_of_Self-Adjoint_Subset_of_Banach_*-Algebra_is_Self-Adjoint | https://proofwiki.org/wiki/Closure_of_Self-Adjoint_Subset_of_Banach_*-Algebra_is_Self-Adjoint | [
"Banach *-Algebras"
] | [
"Definition:*-Algebra",
"Definition:Self-Adjoint Subset of *-Algebra",
"Definition:Closure (Topology)",
"Definition:Self-Adjoint Subset of *-Algebra"
] | [
"Point in Closure of Subset of Metric Space iff Limit of Sequence",
"Definition:Sequence",
"Definition:Self-Adjoint Subset of *-Algebra",
"Star of Convergent Sequence in Banach *-Algebra Converges",
"Point in Closure of Subset of Metric Space iff Limit of Sequence",
"Definition:Self-Adjoint Subset of *-Al... |
proofwiki-22080 | C*-Algebra Generated by Commutative Self-Adjoint Set is Commutative | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $S \subseteq A$ be a self-adjoint set such that:
:$x y = y x$ for each $x, y \in S$.
Let $\map {\text C^\ast} S$ be the $\text C^\ast$-algebra generated by $S$.
Then $\map {\text C^\ast} S$ is commutative. | Let $C$ be the subalgebra generated by $S$.
From Subalgebra Generated by Commuting Elements is Commutative, $C$ is commutative.
From Explicit Form for Generated C*-Algebra, we have $\map {\text C^\ast} S = C^-$.
From Closure of Commutative Set in Banach Algebra is Commutative, $C^-$ is commutative.
Hence $\map {\text... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $S \subseteq A$ be a [[Definition:Self-Adjoint Subset of *-Algebra|self-adjoint set]] such that:
:$x y = y x$ for each $x, y \in S$.
Let $\map {\text C^\ast} S$ be the [[Definition:Generated C*-Algebra|$\text C^\... | Let $C$ be the [[Definition:Generated Subalgebra|subalgebra generated by $S$]].
From [[Subalgebra Generated by Commuting Elements is Commutative]], $C$ is [[Definition:Commutative Algebra (Abstract Algebra)|commutative]].
From [[Explicit Form for Generated C*-Algebra]], we have $\map {\text C^\ast} S = C^-$.
From ... | C*-Algebra Generated by Commutative Self-Adjoint Set is Commutative | https://proofwiki.org/wiki/C*-Algebra_Generated_by_Commutative_Self-Adjoint_Set_is_Commutative | https://proofwiki.org/wiki/C*-Algebra_Generated_by_Commutative_Self-Adjoint_Set_is_Commutative | [
"C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Self-Adjoint Subset of *-Algebra",
"Definition:Generated C*-Algebra",
"Definition:Commutative Algebra (Abstract Algebra)"
] | [
"Definition:Generator of Algebra",
"Subalgebra Generated by Commuting Elements is Commutative",
"Definition:Commutative Algebra (Abstract Algebra)",
"Explicit Form for Generated C*-Algebra",
"Closure of Commutative Set in Banach Algebra is Commutative",
"Definition:Commutative Algebra (Abstract Algebra)",... |
proofwiki-22081 | Spectrum of Hermitian Element in Unital C*-Algebra is Real | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $x \in A$ be Hermitian.
Let $\map {\sigma_A} x$ denote the spectrum of $x$ in $A$.
Then $\map {\sigma_A} x \subseteq \R$. | 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*-Algebra, we have:
:$\map {\sigm... | 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]].
Let $\map {\sigma_A} x$ denote the [[Definition:Spectrum (Spectral Theory)/Unital Algebra|spectrum... | 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 $\Phi_{A'}$ be the [[Definition:Sp... | Spectrum of Hermitian Element in Unital C*-Algebra is Real | https://proofwiki.org/wiki/Spectrum_of_Hermitian_Element_in_Unital_C*-Algebra_is_Real | https://proofwiki.org/wiki/Spectrum_of_Hermitian_Element_in_Unital_C*-Algebra_is_Real | [
"C*-Algebras",
"Hermitian Elements of *-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Hermitian Element of *-Algebra",
"Definition:Spectrum (Spectral Theory)/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-22082 | Spectrum of Element of Unital C*-Subalgebra of Unital C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra with identity element ${\mathbf 1}_A$.
Let $B \subseteq A$ be a unital $\text C^\ast$-subalgebra of $A$.
Let $\sigma_A$ and $\sigma_B$ be the spectrum in $A$ and $B$ respectively.
Let $x \in B$.
Then we have:
:$\map {\sigma_A} x = \map ... | First take $x$ to be Hermitian.
Let $B' \subseteq B$ be the $\text C^\ast$-algebra generated by $\set { {\mathbf 1}_A, x}$.
By C*-Algebra Generated by Commutative Self-Adjoint Set is Commutative, $B'$ is commutative.
From Spectrum of Hermitian Element in Unital C*-Algebra is Real, we have $\map {\sigma_{B'} } x \subs... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]] with [[Definition:Identity Element|identity element]] ${\mathbf 1}_A$.
Let $B \subseteq A$ be a [[Definition:Unital Subalgebra|unital]] [[Definition:C*-Subalgebra|$\text C^\as... | First take $x$ to be [[Definition:Hermitian Element of *-Algebra|Hermitian]].
Let $B' \subseteq B$ 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]], $B'$ is [[Definition:Commutative A... | Spectrum of Element of Unital C*-Subalgebra of Unital C*-Algebra | https://proofwiki.org/wiki/Spectrum_of_Element_of_Unital_C*-Subalgebra_of_Unital_C*-Algebra | https://proofwiki.org/wiki/Spectrum_of_Element_of_Unital_C*-Subalgebra_of_Unital_C*-Algebra | [
"C*-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Unital Subalgebra",
"Definition:C*-Subalgebra",
"Definition:Spectrum (Spectral Theory)/Unital Algebra"
] | [
"Definition:Hermitian Element of *-Algebra",
"Definition:Generated C*-Algebra",
"C*-Algebra Generated by Commutative Self-Adjoint Set is Commutative",
"Definition:Commutative Algebra (Abstract Algebra)",
"Spectrum of Hermitian Element in Unital C*-Algebra is Real",
"Spectrum of Element in Unital Subalgebr... |
proofwiki-22083 | Complete Boolean Lattice is a Frame | Let $\struct{B, \vee, \wedge, \neg, \preceq}$ be a complete Boolean lattice.
Then:
:$\struct{B, \vee, \wedge, \preceq}$ is a frame. | From Boolean Lattice is Heyting Lattice:
:$\struct{B, \vee, \wedge, \preceq}$ is a complete Heyting lattice.
From Characterization of Locale:
:$\struct{B, \vee, \wedge, \preceq}$ is a frame.
{{qed}}
Category:Boolean Lattices
Category:Frames
jf74ombbcrepcbhnlt4bh7wdg3iuybs | Let $\struct{B, \vee, \wedge, \neg, \preceq}$ be a [[Definition:Complete Lattice|complete]] [[Definition:Boolean Lattice|Boolean lattice]].
Then:
:$\struct{B, \vee, \wedge, \preceq}$ is a [[Definition:Frame (Lattice Theory)|frame]]. | From [[Boolean Lattice is Heyting Lattice]]:
:$\struct{B, \vee, \wedge, \preceq}$ is a [[Definition:Complete Lattice|complete]] [[Definition:Heyting Lattice|Heyting lattice]].
From [[Characterization of Locale]]:
:$\struct{B, \vee, \wedge, \preceq}$ is a [[Definition:Frame (Lattice Theory)|frame]].
{{qed}}
[[Categor... | Complete Boolean Lattice is a Frame | https://proofwiki.org/wiki/Complete_Boolean_Lattice_is_a_Frame | https://proofwiki.org/wiki/Complete_Boolean_Lattice_is_a_Frame | [
"Boolean Lattices",
"Frames"
] | [
"Definition:Complete Lattice",
"Definition:Boolean Lattice",
"Definition:Frame (Lattice Theory)"
] | [
"Boolean Lattice is Heyting Lattice",
"Definition:Complete Lattice",
"Definition:Heyting Algebra",
"Characterization of Locale",
"Definition:Frame (Lattice Theory)",
"Category:Boolean Lattices",
"Category:Frames"
] |
proofwiki-22084 | Complete Boolean Lattice is a Locale | Let $\struct{B, \vee, \wedge, \neg, \preceq}$ be a complete Boolean lattice.
Then:
:$\struct{B, \vee, \wedge, \preceq}$ is a locale. | From Boolean Lattice is Heyting Lattice:
:$\struct{B, \vee, \wedge, \preceq}$ is a complete Heyting lattice.
From Characterization of Locale:
:$\struct{B, \vee, \wedge, \preceq}$ is a locale.
{{qed}}
Category:Boolean Lattices
Category:Locales
1d736s6333h98zpxue779ro7pbauzn4 | Let $\struct{B, \vee, \wedge, \neg, \preceq}$ be a [[Definition:Complete Lattice|complete]] [[Definition:Boolean Lattice|Boolean lattice]].
Then:
:$\struct{B, \vee, \wedge, \preceq}$ is a [[Definition:Locale (Lattice Theory)|locale]]. | From [[Boolean Lattice is Heyting Lattice]]:
:$\struct{B, \vee, \wedge, \preceq}$ is a [[Definition:Complete Lattice|complete]] [[Definition:Heyting Lattice|Heyting lattice]].
From [[Characterization of Locale]]:
:$\struct{B, \vee, \wedge, \preceq}$ is a [[Definition:Locale (Lattice Theory)|locale]].
{{qed}}
[[Categ... | Complete Boolean Lattice is a Locale | https://proofwiki.org/wiki/Complete_Boolean_Lattice_is_a_Locale | https://proofwiki.org/wiki/Complete_Boolean_Lattice_is_a_Locale | [
"Boolean Lattices",
"Locales"
] | [
"Definition:Complete Lattice",
"Definition:Boolean Lattice",
"Definition:Locale (Lattice Theory)"
] | [
"Boolean Lattice is Heyting Lattice",
"Definition:Complete Lattice",
"Definition:Heyting Algebra",
"Characterization of Locale",
"Definition:Locale (Lattice Theory)",
"Category:Boolean Lattices",
"Category:Locales"
] |
proofwiki-22085 | Finite Boolean Lattice is a Locale | Let $\struct{B, \vee, \wedge, \neg, \preceq}$ be a finite Boolean lattice.
Then:
:$\struct{B, \vee, \wedge, \preceq}$ is a locale. | By definition of a lattice:
:$B$ admits all finite non-empty suprema and finite non-empty infima.
By definition of a Boolean lattice:
:$B$ has a greatest element and a smallest element.
From Infimum of Empty Set is Greatest Element and Supremum of Empty Set is Smallest Element:
:$B$ admits empty supremum and empty infi... | Let $\struct{B, \vee, \wedge, \neg, \preceq}$ be a [[Definition:Finite Set|finite]] [[Definition:Boolean Lattice|Boolean lattice]].
Then:
:$\struct{B, \vee, \wedge, \preceq}$ is a [[Definition:Locale (Lattice Theory)|locale]]. | By definition of a [[Definition:Lattice (Order Theory)|lattice]]:
:$B$ admits all [[Definition:Finite Supremum|finite]] [[Definition:Empty Supremum|non-empty suprema]] and [[Definition:Finite Infimum|finite]] [[Definition:Empty Infimum|non-empty infima]].
By definition of a [[Definition:Boolean Lattice|Boolean lattic... | Finite Boolean Lattice is a Locale | https://proofwiki.org/wiki/Finite_Boolean_Lattice_is_a_Locale | https://proofwiki.org/wiki/Finite_Boolean_Lattice_is_a_Locale | [
"Boolean Lattices",
"Locales"
] | [
"Definition:Finite Set",
"Definition:Boolean Lattice",
"Definition:Locale (Lattice Theory)"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Supremum of Set/Finite Supremum",
"Definition:Empty Supremum",
"Definition:Infimum of Set/Finite Infimum",
"Definition:Empty Infimum",
"Definition:Boolean Lattice",
"Definition:Greatest Element",
"Definition:Smallest Element",
"Infimum of Empty Set is... |
proofwiki-22086 | Two is Boolean Lattice | Let $\struct{\mathbf 2, \vee, \wedge, \neg, \preceq}$ be (Boolean lattice) two.
Then:
:$\struct{\mathbf 2, \vee, \wedge, \neg, \preceq}$ is a Boolean lattice. | By definition of Boolean lattice two:
:$\struct{\mathbf 2, \vee, \wedge, \neg}$ is Boolean algebra two
From Two is Boolean Algebra:
:$\struct{\mathbf 2, \vee, \wedge, \neg}$ is a Boolean algebra
From Boolean Algebra is Equivalent to Boolean Lattice:
:$\struct{\mathbf 2, \vee, \wedge, \neg, \preccurlyeq}$ is a Boolean l... | Let $\struct{\mathbf 2, \vee, \wedge, \neg, \preceq}$ be [[Definition:Two (Boolean Lattice)|(Boolean lattice) two]].
Then:
:$\struct{\mathbf 2, \vee, \wedge, \neg, \preceq}$ is a [[Definition:Boolean Lattice|Boolean lattice]]. | By definition of [[Definition:Two (Boolean Lattice)|Boolean lattice two]]:
:$\struct{\mathbf 2, \vee, \wedge, \neg}$ is [[Definition:Two (Boolean Algebra)|Boolean algebra two]]
From [[Two is Boolean Algebra]]:
:$\struct{\mathbf 2, \vee, \wedge, \neg}$ is a [[Definition:Boolean Algebra|Boolean algebra]]
From [[Boole... | Two is Boolean Lattice | https://proofwiki.org/wiki/Two_is_Boolean_Lattice | https://proofwiki.org/wiki/Two_is_Boolean_Lattice | [
"Boolean Lattices"
] | [
"Definition:Two (Boolean Lattice)",
"Definition:Boolean Lattice"
] | [
"Definition:Two (Boolean Lattice)",
"Definition:Two (Boolean Algebra)",
"Two is Boolean Algebra",
"Definition:Boolean Algebra",
"Boolean Algebra is Equivalent to Boolean Lattice",
"Definition:Boolean Lattice",
"Definition:Two (Boolean Algebra)",
"Definition:Cayley Table",
"Definition:Two (Boolean La... |
proofwiki-22087 | Two is a Locale | Let $\struct{\mathbf 2, \vee, \wedge, \neg, \preceq}$ denote (Boolean lattice) two.
Then:
:$\struct{\mathbf 2, \vee, \wedge, \preceq}$ is a locale. | Follows immediately from:
* Two is Boolean Lattice
* Finite Boolean Lattice is a Locale
{{qed}}
Category:Boolean Lattices
Category:Locales
sucnmkknv7ukc9n7odqg53odogbhwga | Let $\struct{\mathbf 2, \vee, \wedge, \neg, \preceq}$ denote [[Definition:Two (Boolean Lattice)|(Boolean lattice) two]].
Then:
:$\struct{\mathbf 2, \vee, \wedge, \preceq}$ is a [[Definition:Locale (Lattice Theory)|locale]]. | Follows immediately from:
* [[Two is Boolean Lattice]]
* [[Finite Boolean Lattice is a Locale]]
{{qed}}
[[Category:Boolean Lattices]]
[[Category:Locales]]
sucnmkknv7ukc9n7odqg53odogbhwga | Two is a Locale | https://proofwiki.org/wiki/Two_is_a_Locale | https://proofwiki.org/wiki/Two_is_a_Locale | [
"Boolean Lattices",
"Locales"
] | [
"Definition:Two (Boolean Lattice)",
"Definition:Locale (Lattice Theory)"
] | [
"Two is Boolean Lattice",
"Finite Boolean Lattice is a Locale",
"Category:Boolean Lattices",
"Category:Locales"
] |
proofwiki-22088 | Boolean Lattice is Heyting Lattice | Let $\struct{B, \vee, \wedge, \neg, \preceq}$ be a Boolean lattice.
Then:
:$\struct{B, \vee, \wedge, \preceq}$ is a Heyting lattice where:
::$\forall x, y \in B : x \to y = \neg x \vee y$
:and:
::$x \to y$ denotes the relative pseudocomplement of $x$ with respect to $y$ | By definition of Boolean lattice and Heyting lattice it remains to show that for all $x, y \in B$ the relative pseudocomplement of $x$ with respect to $y$ exists.
It will be shown that:
:$\forall x, y \in B : \neg x \vee y$ is the relative pseudocomplement of $x$ with respect to $y$
By definition of relative pseudocomp... | Let $\struct{B, \vee, \wedge, \neg, \preceq}$ be a [[Definition:Boolean Lattice|Boolean lattice]].
Then:
:$\struct{B, \vee, \wedge, \preceq}$ is a [[Definition:Heyting Lattice|Heyting lattice]] where:
::$\forall x, y \in B : x \to y = \neg x \vee y$
:and:
::$x \to y$ denotes the [[Definition:Relative Pseudocomplement... | By definition of [[Definition:Boolean Lattice|Boolean lattice]] and [[Definition:Heyting Lattice|Heyting lattice]] it remains to show that for all $x, y \in B$ the [[Definition:Relative Pseudocomplement|relative pseudocomplement of $x$ with respect to $y$]] exists.
It will be shown that:
:$\forall x, y \in B : \neg x... | Boolean Lattice is Heyting Lattice | https://proofwiki.org/wiki/Boolean_Lattice_is_Heyting_Lattice | https://proofwiki.org/wiki/Boolean_Lattice_is_Heyting_Lattice | [
"Boolean Lattices",
"Heyting Algebras"
] | [
"Definition:Boolean Lattice",
"Definition:Heyting Algebra",
"Definition:Relative Pseudocomplement"
] | [
"Definition:Boolean Lattice",
"Definition:Heyting Algebra",
"Definition:Relative Pseudocomplement",
"Definition:Relative Pseudocomplement",
"Definition:Relative Pseudocomplement",
"Meet Semilattice is Ordered Structure",
"Definition:Boolean Lattice",
"Definition:Distributive Lattice",
"Join Semilatt... |
proofwiki-22089 | Odd Function/Examples/x^3 | Let $f: \R \to \R$ denote the cube function on $\R$.
:$\forall x \in \R: \map f x = x^3$
Then $f$ is an odd function. | {{ProofWanted|A specific instance of Odd Power Function is Odd which bizarrely we don't have}} | Let $f: \R \to \R$ denote the [[Definition:Cube (Algebra)|cube function]] on $\R$.
:$\forall x \in \R: \map f x = x^3$
Then $f$ is an [[Definition:Odd Function|odd function]]. | {{ProofWanted|A specific instance of [[Odd Power Function is Odd]] which bizarrely we don't have}} | Odd Function/Examples/x^3 | https://proofwiki.org/wiki/Odd_Function/Examples/x^3 | https://proofwiki.org/wiki/Odd_Function/Examples/x^3 | [
"Cube Function",
"Examples of Odd Functions"
] | [
"Definition:Cube/Algebra",
"Definition:Odd Function"
] | [
"Odd Power Function is Odd"
] |
proofwiki-22090 | Equivalence of Definitions of Odd Permutation | {{TFAE|def = Odd Permutation}}
Let $n \in \N$ be a natural number.
Let $S_n$ denote the symmetric group on $n$ letters.
Let $\rho \in S_n$ be a permutation in $S_n$. | The '''sign of $\rho$''' is defined as:
:$\map \sgn \rho = \begin {cases} 1 & : \text {$k$ even} \\ -1 & : \text {$k$ odd} \\ \end {cases}$
The result follows.
{{qed}}
Category:Odd Permutations
ftgvi42cd03vppbmaetmhg0qenqs5ol | {{TFAE|def = Odd Permutation}}
Let $n \in \N$ be a [[Definition:Natural Number|natural number]].
Let $S_n$ denote the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]].
Let $\rho \in S_n$ be a [[Definition:Permutation on n Letters|permutation in $S_n$]]. | The '''[[Definition:Sign of Permutation on n Letters|sign of $\rho$]]''' is defined as:
:$\map \sgn \rho = \begin {cases} 1 & : \text {$k$ even} \\ -1 & : \text {$k$ odd} \\ \end {cases}$
The result follows.
{{qed}}
[[Category:Odd Permutations]]
ftgvi42cd03vppbmaetmhg0qenqs5ol | Equivalence of Definitions of Odd Permutation | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Odd_Permutation | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Odd_Permutation | [
"Odd Permutations"
] | [
"Definition:Natural Numbers",
"Definition:Symmetric Group/n Letters",
"Definition:Permutation on n Letters"
] | [
"Definition:Sign of Permutation on n Letters",
"Category:Odd Permutations"
] |
proofwiki-22091 | Characterization of Homeomorphic Topological Spaces | Let $T_1 = \struct{S_1, \tau_1}$ be topological space.
Let $S_2$ be a set.
Let $\tau_2$ be a subset of the powerset $\powerset {S_2}$.
Then:
:$\struct{S_2, \tau_2}$ is a topological space homeomorphic to $T_1$
{{iff}}:
:there exists a mapping $f : S_1 \to S_2$:
::$(1)\quad f$ is a bijection
::$(2)\quad f^\to \restricti... | === Necessary Condition ===
Let $\struct{S_2, \tau_2}$ be a topological space homeomorphic to $T_1$.
{{:Characterization of Homeomorphic Topological Spaces/Necessary Condition}}{{qed|lemma}} | Let $T_1 = \struct{S_1, \tau_1}$ be [[Definition:Topological Space|topological space]].
Let $S_2$ be a [[Definition:Set|set]].
Let $\tau_2$ be a [[Definition:Subset|subset]] of the [[Definition:Powerset|powerset]] $\powerset {S_2}$.
Then:
:$\struct{S_2, \tau_2}$ is a [[Definition:Topological Space|topological spac... | === [[Characterization of Homeomorphic Topological Spaces/Necessary Condition|Necessary Condition]] ===
Let $\struct{S_2, \tau_2}$ be a [[Definition:Topological Space|topological space]] [[Definition:Homeomorphic Topological Spaces|homeomorphic]] to $T_1$.
{{:Characterization of Homeomorphic Topological Spaces/Necessa... | Characterization of Homeomorphic Topological Spaces | https://proofwiki.org/wiki/Characterization_of_Homeomorphic_Topological_Spaces | https://proofwiki.org/wiki/Characterization_of_Homeomorphic_Topological_Spaces | [
"Homeomorphisms (Topological Spaces)",
"Characterization of Homeomorphic Topological Spaces"
] | [
"Definition:Topological Space",
"Definition:Set",
"Definition:Subset",
"Definition:Power Set",
"Definition:Topological Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Mapping",
"Definition:Bijection",
"Definition:Surjection",
"Definition:Restriction/Mapping",
"Definition:Direc... | [
"Characterization of Homeomorphic Topological Spaces/Necessary Condition",
"Definition:Topological Space",
"Definition:Homeomorphism/Topological Spaces"
] |
proofwiki-22092 | Spectrum of Locale is Sober Space | Let $\struct{L, \preceq}$ be a locale.
Let $\map {\operatorname{Sp}} L$ denote the spectrum of $L$.
Then:
:$\map {\operatorname{Sp}} L$ is a sober space. | Let $\map {\operatorname{Sp}} L$ denote the spectrum of $L$ as completely prime filters.
From Spectrum of Locale as Completely Prime Filters is Sober Space:
:$\map {\operatorname{Sp}} L$ is a sober space
From Homeomorphism Preserves Sobriety:
:every topological space homeomorphic to $\map {\operatorname{Sp}} L$ is a so... | Let $\struct{L, \preceq}$ be a [[Definition:Locale (Lattice Theory)|locale]].
Let $\map {\operatorname{Sp}} L$ denote the [[Definition:Spectrum of Locale|spectrum]] of $L$.
Then:
:$\map {\operatorname{Sp}} L$ is a [[Definition:Sober Space|sober space]]. | Let $\map {\operatorname{Sp}} L$ denote the [[Definition:Spectrum of Locale as Completely Prime Filters|spectrum of $L$ as completely prime filters]].
From [[Spectrum of Locale as Completely Prime Filters is Sober Space]]:
:$\map {\operatorname{Sp}} L$ is a [[Definition:Sober Space|sober space]]
From [[Homeomorphis... | Spectrum of Locale is Sober Space | https://proofwiki.org/wiki/Spectrum_of_Locale_is_Sober_Space | https://proofwiki.org/wiki/Spectrum_of_Locale_is_Sober_Space | [
"Spectra of Locales"
] | [
"Definition:Locale (Lattice Theory)",
"Definition:Spectrum of Locale",
"Definition:Sober Space"
] | [
"Definition:Spectrum of Locale/Completely Prime Filters",
"Spectrum of Locale as Completely Prime Filters is Sober Space",
"Definition:Sober Space",
"Homeomorphism Preserves Sobriety",
"Definition:Topological Space",
"Definition:Homeomorphism",
"Definition:Sober Space",
"Topological Equivalence of Def... |
proofwiki-22093 | Topological Equivalence of Definitions of Spectrum of Locale | Let $\struct{L, \preceq}$ be a locale.
{{TFAETop|def = Spectrum of Locale|view = spectrum of locale}} | From
:* Spectrum of Locale as Completely Prime Filters is Sober Space:
the spectrum of $L$ as completely prime filters is a sober space.
From:
:* Canonical Bijection from Completely Prime Filters to Frame Homomorphisms
:* Characterization of Homeomorphic Topological Spaces
the spectrum of $L$ as frame homomorphisms is... | Let $\struct{L, \preceq}$ be a [[Definition:Locale (Lattice Theory)|locale]].
{{TFAETop|def = Spectrum of Locale|view = spectrum of locale}} | From
:* [[Spectrum of Locale as Completely Prime Filters is Sober Space]]:
the [[Definition:Spectrum of Locale as Completely Prime Filters|spectrum of $L$ as completely prime filters]] is a [[Definition:Sober Space|sober space]].
From:
:* [[Canonical Bijection from Completely Prime Filters to Frame Homomorphisms]]
:... | Topological Equivalence of Definitions of Spectrum of Locale | https://proofwiki.org/wiki/Topological_Equivalence_of_Definitions_of_Spectrum_of_Locale | https://proofwiki.org/wiki/Topological_Equivalence_of_Definitions_of_Spectrum_of_Locale | [
"Spectra of Locales"
] | [
"Definition:Locale (Lattice Theory)"
] | [
"Spectrum of Locale as Completely Prime Filters is Sober Space",
"Definition:Spectrum of Locale/Completely Prime Filters",
"Definition:Sober Space",
"Canonical Bijection from Completely Prime Filters to Frame Homomorphisms",
"Characterization of Homeomorphic Topological Spaces",
"Definition:Spectrum of Lo... |
proofwiki-22094 | Spectrum Functor is Covariant Functor | Let $\mathbf{Loc}$ denote the category of locales.
Let $\mathbf{Top}$ denote the category of topological spaces.
Then:
:the spectrum functor $\operatorname {Sp} : \mathbf{Loc} \to \mathbf{Top}$ is a covariant functor | Recall that the category of locales $\mathbf{Loc}$ is the dual category of the category of frames $\mathbf{Frm}$ by definition.
For each continuous map $f: L_1 \to L_2$ in $\mathbf{Loc}$, let:
:$f^*: L_2 \to L_1$ denote the frame homomorphism in $\mathbf{Frm}$ such that $f = \paren{f^*}^{\text{op} }$.
Let $\operatornam... | Let $\mathbf{Loc}$ denote the [[Definition:Category of Locales|category of locales]].
Let $\mathbf{Top}$ denote the [[Definition:Category of Topological Spaces|category of topological spaces]].
Then:
:the [[Definition:Spectrum Functor|spectrum functor]] $\operatorname {Sp} : \mathbf{Loc} \to \mathbf{Top}$ is a [[Def... | Recall that the [[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]] $\mathbf{Frm}$ by definition.
For each [[Definition:Continuous Map (Locale)|continuous map]] $f: L_1 \to L_2$ in $\mathbf{... | Spectrum Functor is Covariant Functor | https://proofwiki.org/wiki/Spectrum_Functor_is_Covariant_Functor | https://proofwiki.org/wiki/Spectrum_Functor_is_Covariant_Functor | [
"Functors"
] | [
"Definition:Category of Locales",
"Definition:Category of Topological Spaces",
"Definition:Spectrum Functor",
"Definition:Functor/Covariant"
] | [
"Definition:Category of Locales",
"Definition:Dual Category",
"Definition:Category of Frames",
"Definition:Continuous Map (Locale)",
"Definition:Frame Homomorphism",
"Definition:Frame Spectrum Functor",
"Frame Spectrum Functor is Contravariant Functor",
"Definition:Functor/Contravariant",
"Contravar... |
proofwiki-22095 | Character on Unital C*-Algebra has Modulus One at Unitary Elements | Let $\tuple {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $\phi$ be a character on $A$.
Let $x \in A$ be unitary.
Then:
:$\cmod {\map \phi x} = 1$ | Let ${\mathbf 1}_A$ be the identity element of $A$.
Since $x$ is unitary, we have:
:$x x^\ast = {\mathbf 1}_A$
Hence:
:$\map \phi {x x^\ast} = \map \phi { {\mathbf 1}_A}$
From Character on C*-Algebra is *-Algebra Homomorphism, we have:
:$\map \phi {x x^\ast} = \map \phi x \map \phi {x^\ast} = \map \phi x \overline {\m... | Let $\tuple {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\phi$ be a [[Definition:Character (Banach Algebra)|character]] on $A$.
Let $x \in A$ be [[Definition:Unitary Element of Unital *-Algebra|unitary]].
Then:
:$\cmod {\map... | Let ${\mathbf 1}_A$ be the [[Definition:Identity Element|identity element]] of $A$.
Since $x$ is [[Definition:Unitary Element of Unital *-Algebra|unitary]], we have:
:$x x^\ast = {\mathbf 1}_A$
Hence:
:$\map \phi {x x^\ast} = \map \phi { {\mathbf 1}_A}$
From [[Character on C*-Algebra is *-Algebra Homomorphism]], we... | Character on Unital C*-Algebra has Modulus One at Unitary Elements | https://proofwiki.org/wiki/Character_on_Unital_C*-Algebra_has_Modulus_One_at_Unitary_Elements | https://proofwiki.org/wiki/Character_on_Unital_C*-Algebra_has_Modulus_One_at_Unitary_Elements | [
"Characters (Banach Algebras)",
"C*-Algebras",
"Unitary Elements of Unital *-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Character (Banach Algebra)",
"Definition:Unitary Element of Unital *-Algebra"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Unitary Element of Unital *-Algebra",
"Character on C*-Algebra is *-Algebra Homomorphism",
"Product of Complex Number with Conjugate",
"Character on Unital Banach Algebra is Unital Algebra Homomorphism",
"Category:Characters (Banach ... |
proofwiki-22096 | Spectrum of Unitary Element in Unital C*-Algebra is Subset of Unit Circle | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $x \in A$ be unitary.
Let $\map {\sigma_A} x$ denote the spectrum of $x$ in $A$.
Then $\map {\sigma_A} x \subseteq \set {z \in \C : \cmod z = 1}$. | Let $A' \subseteq A$ be the $\text C^\ast$-algebra generated by $\set { {\mathbf 1}_A, x, x^\ast}$.
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*-Algebra, we have:
:$\ma... | 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:Unitary Element of Unital *-Algebra|unitary]].
Let $\map {\sigma_A} x$ denote the [[Definition:Spectrum (Spectral Theory)/Unital Algebra|spect... | Let $A' \subseteq A$ be the [[Definition:Generated C*-Algebra|$\text C^\ast$-algebra generated]] by $\set { {\mathbf 1}_A, x, x^\ast}$.
By [[C*-Algebra Generated by Commutative Self-Adjoint Set is Commutative]], $A'$ is [[Definition:Commutative Algebra (Abstract Algebra)|commutative]].
Let $\Phi_{A'}$ be the [[Defin... | Spectrum of Unitary Element in Unital C*-Algebra is Subset of Unit Circle | https://proofwiki.org/wiki/Spectrum_of_Unitary_Element_in_Unital_C*-Algebra_is_Subset_of_Unit_Circle | https://proofwiki.org/wiki/Spectrum_of_Unitary_Element_in_Unital_C*-Algebra_is_Subset_of_Unit_Circle | [
"C*-Algebras",
"Unitary Elements of Unital *-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Unitary Element of Unital *-Algebra",
"Definition:Spectrum (Spectral Theory)/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-22097 | Existence and Uniqueness of Continuous Functional Calculus | 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 $\iota : \map {\sigma_A} x \to \C$ be the inclusion mapping.
Then there exists a unique unital $\ast$-algebra homomorphism $\Theta_x : \map \CC {\map... | Let $B \subseteq A$ be the $\text C^\ast$-algebra generated by $\set { {\mathbf 1}_A, x, x^\ast}$.
By C*-Algebra Generated by Commutative Self-Adjoint Set is Commutative, $B$ is commutative.
From Spectrum of Element of Unital C*-Subalgebra of Unital C*-Algebra, we have:
:$\map {\sigma_A} x = \map {\sigma_B} x$
Let $\P... | 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 $\set { {\mathbf 1}_A, x, x^\ast}$.
By [[C*-Algebra Generated by Commutative Self-Adjoint Set is Commutative]], $B$ is [[Definition:Commutative Algebra (Abstract Algebra)|commutative]].
From [[Spectrum of Element of Un... | Existence and Uniqueness of Continuous Functional Calculus | https://proofwiki.org/wiki/Existence_and_Uniqueness_of_Continuous_Functional_Calculus | https://proofwiki.org/wiki/Existence_and_Uniqueness_of_Continuous_Functional_Calculus | [
"Continuous Functional Calculus",
"C*-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Normal Element of *-Algebra",
"Definition:Spectrum (Spectral Theory)/Unital Algebra",
"Definition:Inclusion Mapping",
"Definition:Unital *-Algebra Homomorphism",
"Definition:Space of Continuous Functions on Compact Hausdorff Space"... | [
"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",
"Definition:Gelfand Transform",
"Gel... |
proofwiki-22098 | Continuous Mapping Induced by Continuous Map is Continuous | Let $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ be locales.
Let $f : L_1 \to L_2$ be a continuous map.
Let:
:$\map {\operatorname{Sp}} f : \map {\operatorname{Sp}} {L_1} \to \map {\operatorname{Sp}} {L_2}$ denote the continuous mapping induced by $f$
where:
:$\map {\operatorname{Sp}} {L_1}$ and ... | Let $f^* : L_2 \to L_1$ denote the frame homomorphism in the category of frames $\mathbf {Frm}$ such that:
:$f = \paren{f^*}^{\text{op} }$ in the category of locales $\mathbf {Loc}$.
Let:
:$\map {\operatorname{Sp}} {f^*} : \map {\operatorname{Sp}} {L_1} \to \map {\operatorname{Sp}} {L_2}$ denote the continuous mapping ... | Let $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ be [[Definition:Locale (Lattice Theory)|locales]].
Let $f : L_1 \to L_2$ be a [[Definition:Continuous Map (Locale)|continuous map]].
Let:
:$\map {\operatorname{Sp}} f : \map {\operatorname{Sp}} {L_1} \to \map {\operatorname{Sp}} {L_2}$ denote th... | Let $f^* : L_2 \to L_1$ denote the [[Definition:Frame Homomorphism|frame homomorphism]] in the [[Definition:Category of Frames|category of frames]] $\mathbf {Frm}$ such that:
:$f = \paren{f^*}^{\text{op} }$ in the [[Definition:Category of Locales|category of locales]] $\mathbf {Loc}$.
Let:
:$\map {\operatorname{Sp}} ... | Continuous Mapping Induced by Continuous Map is Continuous | https://proofwiki.org/wiki/Continuous_Mapping_Induced_by_Continuous_Map_is_Continuous | https://proofwiki.org/wiki/Continuous_Mapping_Induced_by_Continuous_Map_is_Continuous | [
"Continuous Maps",
"Continuous Mappings"
] | [
"Definition:Locale (Lattice Theory)",
"Definition:Continuous Map (Locale)",
"Definition:Continuous Mapping Induced by Continuous Map",
"Definition:Spectrum of Locale/Completely Prime Filters",
"Definition:Continuous Mapping (Topology)/Everywhere"
] | [
"Definition:Frame Homomorphism",
"Definition:Category of Frames",
"Definition:Category of Locales",
"Definition:Continuous Mapping Induced by Frame Homomorphism",
"Continuous Mapping Induced by Frame Homomorphism is Continuous",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Continuo... |
proofwiki-22099 | Spaces of Continuous Functions on Homeomorphic Compact Hausdorff Spaces are Isometrically *-Algebra Isomorphic | Let $K, L$ be homeomorphic compact Hausdorff spaces.
Let $\map \CC K$ and $\map \CC L$ be the spaces of continuous functions on $K$ and $L$ respectively.
Then $\map \CC K$ and $\map \CC L$ are isometrically $\ast$-algebra isomorphic. | Let $f : K \to L$ be a homeomorphism.
Let $\phi \in \map \CC L$.
From Composite of Continuous Mappings is Continuous, $\phi \circ f \in \map \CC K$.
Hence we can define $T : \map \CC L \to \map \CC K$ by:
:$T \phi = \phi \circ f$
for each $\phi \in \map \CC L$.
Let $\phi, \psi \in \map \CC L$, $\lambda \in \C$ and $x ... | Let $K, L$ be [[Definition:Homeomorphism|homeomorphic]] [[Definition:Compact Topological Space|compact]] [[Definition:Hausdorff Space|Hausdorff spaces]].
Let $\map \CC K$ and $\map \CC L$ be the [[Definition:Space of Continuous Functions on Compact Hausdorff Space|spaces of continuous functions]] on $K$ and $L$ respec... | Let $f : K \to L$ be a [[Definition:Homeomorphism|homeomorphism]].
Let $\phi \in \map \CC L$.
From [[Composite of Continuous Mappings is Continuous]], $\phi \circ f \in \map \CC K$.
Hence we can define $T : \map \CC L \to \map \CC K$ by:
:$T \phi = \phi \circ f$
for each $\phi \in \map \CC L$.
Let $\phi, \psi \in... | Spaces of Continuous Functions on Homeomorphic Compact Hausdorff Spaces are Isometrically *-Algebra Isomorphic | https://proofwiki.org/wiki/Spaces_of_Continuous_Functions_on_Homeomorphic_Compact_Hausdorff_Spaces_are_Isometrically_*-Algebra_Isomorphic | https://proofwiki.org/wiki/Spaces_of_Continuous_Functions_on_Homeomorphic_Compact_Hausdorff_Spaces_are_Isometrically_*-Algebra_Isomorphic | [
"Space of Continuous Functions on Compact Hausdorff Space",
"*-Algebras"
] | [
"Definition:Homeomorphism",
"Definition:Compact Topological Space",
"Definition:T2 Space",
"Definition:Space of Continuous Functions on Compact Hausdorff Space",
"Definition:Linear Isometry",
"Definition:*-Algebra Isomorphism"
] | [
"Definition:Homeomorphism",
"Composite of Continuous Mappings is Continuous",
"Definition:Linear Transformation",
"Definition:Algebra Homomorphism",
"Definition:*-Algebra Homomorphism",
"Definition:Algebra Isomorphism",
"Definition:Bijection",
"Definition:Injection",
"Definition:Continuous Mapping",... |
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