statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
elemental_star_algebra.continuous_character_space_to_spectrum (x : A) :
continuous (elemental_star_algebra.character_space_to_spectrum x) | continuous_induced_rng.2
(map_continuous $ gelfand_transform ℂ (elemental_star_algebra ℂ x) ⟨x, self_mem ℂ x⟩) | lemma | elemental_star_algebra.continuous_character_space_to_spectrum | analysis.normed_space.star | src/analysis/normed_space/star/continuous_functional_calculus.lean | [
"analysis.normed_space.star.gelfand_duality",
"topology.algebra.star_subalgebra"
] | [
"continuous",
"elemental_star_algebra",
"elemental_star_algebra.character_space_to_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
elemental_star_algebra.bijective_character_space_to_spectrum :
function.bijective (elemental_star_algebra.character_space_to_spectrum a) | begin
refine ⟨λ φ ψ h, star_alg_hom_class_ext ℂ (map_continuous φ) (map_continuous ψ)
(by simpa only [elemental_star_algebra.character_space_to_spectrum, subtype.mk_eq_mk,
continuous_map.coe_mk] using h), _⟩,
rintros ⟨z, hz⟩,
have hz' := (star_subalgebra.spectrum_eq (elemental_star_algebra.is_closed ℂ a... | lemma | elemental_star_algebra.bijective_character_space_to_spectrum | analysis.normed_space.star | src/analysis/normed_space/star/continuous_functional_calculus.lean | [
"analysis.normed_space.star.gelfand_duality",
"topology.algebra.star_subalgebra"
] | [
"continuous_map.coe_mk",
"elemental_star_algebra.character_space_to_spectrum",
"elemental_star_algebra.is_closed",
"star_subalgebra.spectrum_eq",
"subtype.mk_eq_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
elemental_star_algebra.character_space_homeo :
character_space ℂ (elemental_star_algebra ℂ a) ≃ₜ spectrum ℂ a | @continuous.homeo_of_equiv_compact_to_t2 _ _ _ _ _ _
(equiv.of_bijective (elemental_star_algebra.character_space_to_spectrum a)
(elemental_star_algebra.bijective_character_space_to_spectrum a))
(elemental_star_algebra.continuous_character_space_to_spectrum a) | def | elemental_star_algebra.character_space_homeo | analysis.normed_space.star | src/analysis/normed_space/star/continuous_functional_calculus.lean | [
"analysis.normed_space.star.gelfand_duality",
"topology.algebra.star_subalgebra"
] | [
"continuous.homeo_of_equiv_compact_to_t2",
"elemental_star_algebra",
"elemental_star_algebra.bijective_character_space_to_spectrum",
"elemental_star_algebra.character_space_to_spectrum",
"elemental_star_algebra.continuous_character_space_to_spectrum",
"equiv.of_bijective",
"spectrum"
] | The homeomorphism between the character space of the unital C⋆-subalgebra generated by a
single normal element `a : A` and `spectrum ℂ a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_functional_calculus :
C(spectrum ℂ a, ℂ) ≃⋆ₐ[ℂ] elemental_star_algebra ℂ a | ((elemental_star_algebra.character_space_homeo a).comp_star_alg_equiv' ℂ ℂ).trans
(gelfand_star_transform (elemental_star_algebra ℂ a)).symm | def | continuous_functional_calculus | analysis.normed_space.star | src/analysis/normed_space/star/continuous_functional_calculus.lean | [
"analysis.normed_space.star.gelfand_duality",
"topology.algebra.star_subalgebra"
] | [
"elemental_star_algebra",
"elemental_star_algebra.character_space_homeo",
"gelfand_star_transform",
"spectrum"
] | **Continuous functional calculus.** Given a normal element `a : A` of a unital C⋆-algebra,
the continuous functional calculus is a `star_alg_equiv` from the complex-valued continuous
functions on the spectrum of `a` to the unital C⋆-subalgebra generated by `a`. Moreover, this
equivalence identifies `(continuous_map.id ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_functional_calculus_map_id :
continuous_functional_calculus a ((continuous_map.id ℂ).restrict (spectrum ℂ a)) =
⟨a, self_mem ℂ a⟩ | star_alg_equiv.symm_apply_apply _ _ | lemma | continuous_functional_calculus_map_id | analysis.normed_space.star | src/analysis/normed_space/star/continuous_functional_calculus.lean | [
"analysis.normed_space.star.gelfand_duality",
"topology.algebra.star_subalgebra"
] | [
"continuous_functional_calculus",
"continuous_map.id",
"spectrum",
"star_alg_equiv.symm_apply_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_adjoint.exp_unitary (a : self_adjoint A) : unitary A | ⟨exp ℂ (I • a), exp_mem_unitary_of_mem_skew_adjoint _ (a.prop.smul_mem_skew_adjoint conj_I)⟩ | def | self_adjoint.exp_unitary | analysis.normed_space.star | src/analysis/normed_space/star/exponential.lean | [
"analysis.normed_space.exponential"
] | [
"exp_mem_unitary_of_mem_skew_adjoint",
"self_adjoint",
"unitary"
] | The map from the selfadjoint real subspace to the unitary group. This map only makes sense
over ℂ. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
commute.exp_unitary_add {a b : self_adjoint A} (h : commute (a : A) (b : A)) :
exp_unitary (a + b) = exp_unitary a * exp_unitary b | begin
ext,
have hcomm : commute (I • (a : A)) (I • (b : A)),
calc _ = _ : by simp only [h.eq, algebra.smul_mul_assoc, algebra.mul_smul_comm],
simpa only [exp_unitary_coe, add_subgroup.coe_add, smul_add] using exp_add_of_commute hcomm,
end | lemma | commute.exp_unitary_add | analysis.normed_space.star | src/analysis/normed_space/star/exponential.lean | [
"analysis.normed_space.exponential"
] | [
"algebra.mul_smul_comm",
"algebra.smul_mul_assoc",
"commute",
"exp_add_of_commute",
"self_adjoint",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.exp_unitary {a b : self_adjoint A} (h : commute (a : A) (b : A)) :
commute (exp_unitary a) (exp_unitary b) | calc (exp_unitary a) * (exp_unitary b) = (exp_unitary b) * (exp_unitary a)
: by rw [←h.exp_unitary_add, ←h.symm.exp_unitary_add, add_comm] | lemma | commute.exp_unitary | analysis.normed_space.star | src/analysis/normed_space/star/exponential.lean | [
"analysis.normed_space.exponential"
] | [
"commute",
"self_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.to_character_space : character_space ℂ A | character_space.equiv_alg_hom.symm $ ((@normed_ring.alg_equiv_complex_of_complete (A ⧸ I) _ _
(by { letI := quotient.field I, exact @is_unit_iff_ne_zero (A ⧸ I) _ }) _).symm :
A ⧸ I →ₐ[ℂ] ℂ).comp
(quotient.mkₐ ℂ I) | def | ideal.to_character_space | analysis.normed_space.star | src/analysis/normed_space/star/gelfand_duality.lean | [
"analysis.normed_space.star.spectrum",
"analysis.normed.group.quotient",
"analysis.normed_space.algebra",
"topology.continuous_function.units",
"topology.continuous_function.compact",
"topology.algebra.algebra",
"topology.continuous_function.stone_weierstrass"
] | [
"is_unit_iff_ne_zero",
"normed_ring.alg_equiv_complex_of_complete"
] | Every maximal ideal in a commutative complex Banach algebra gives rise to a character on that
algebra. In particular, the character, which may be identified as an algebra homomorphism due to
`weak_dual.character_space.equiv_alg_hom`, is given by the composition of the quotient map and
the Gelfand-Mazur isomorphism `nor... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.to_character_space_apply_eq_zero_of_mem {a : A} (ha : a ∈ I) :
I.to_character_space a = 0 | begin
unfold ideal.to_character_space,
simpa only [character_space.equiv_alg_hom_symm_coe, alg_hom.coe_comp,
alg_equiv.coe_alg_hom, quotient.mkₐ_eq_mk, function.comp_app, quotient.eq_zero_iff_mem.mpr ha,
spectrum.zero_eq, normed_ring.alg_equiv_complex_of_complete_symm_apply]
using set.eq_of_mem_singleto... | lemma | ideal.to_character_space_apply_eq_zero_of_mem | analysis.normed_space.star | src/analysis/normed_space/star/gelfand_duality.lean | [
"analysis.normed_space.star.spectrum",
"analysis.normed.group.quotient",
"analysis.normed_space.algebra",
"topology.continuous_function.units",
"topology.continuous_function.compact",
"topology.algebra.algebra",
"topology.continuous_function.stone_weierstrass"
] | [
"alg_equiv.coe_alg_hom",
"alg_hom.coe_comp",
"ideal.to_character_space",
"set.eq_of_mem_singleton",
"set.singleton_nonempty",
"spectrum.zero_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
weak_dual.character_space.exists_apply_eq_zero {a : A} (ha : ¬ is_unit a) :
∃ f : character_space ℂ A, f a = 0 | begin
unfreezingI { obtain ⟨M, hM, haM⟩ := (span {a}).exists_le_maximal (span_singleton_ne_top ha) },
exact ⟨M.to_character_space, M.to_character_space_apply_eq_zero_of_mem
(haM (mem_span_singleton.mpr ⟨1, (mul_one a).symm⟩))⟩,
end | lemma | weak_dual.character_space.exists_apply_eq_zero | analysis.normed_space.star | src/analysis/normed_space/star/gelfand_duality.lean | [
"analysis.normed_space.star.spectrum",
"analysis.normed.group.quotient",
"analysis.normed_space.algebra",
"topology.continuous_function.units",
"topology.continuous_function.compact",
"topology.algebra.algebra",
"topology.continuous_function.stone_weierstrass"
] | [
"is_unit",
"mul_one"
] | If `a : A` is not a unit, then some character takes the value zero at `a`. This is equivlaent
to `gelfand_transform ℂ A a` takes the value zero at some character. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
weak_dual.character_space.mem_spectrum_iff_exists {a : A} {z : ℂ} :
z ∈ spectrum ℂ a ↔ ∃ f : character_space ℂ A, f a = z | begin
refine ⟨λ hz, _, _⟩,
{ obtain ⟨f, hf⟩ := weak_dual.character_space.exists_apply_eq_zero hz,
simp only [map_sub, sub_eq_zero, alg_hom_class.commutes, algebra.id.map_eq_id,
ring_hom.id_apply] at hf,
exact (continuous_map.spectrum_eq_range (gelfand_transform ℂ A a)).symm ▸ ⟨f, hf.symm⟩ },
{ rintr... | lemma | weak_dual.character_space.mem_spectrum_iff_exists | analysis.normed_space.star | src/analysis/normed_space/star/gelfand_duality.lean | [
"analysis.normed_space.star.spectrum",
"analysis.normed.group.quotient",
"analysis.normed_space.algebra",
"topology.continuous_function.units",
"topology.continuous_function.compact",
"topology.algebra.algebra",
"topology.continuous_function.stone_weierstrass"
] | [
"alg_hom.apply_mem_spectrum",
"algebra.id.map_eq_id",
"continuous_map.spectrum_eq_range",
"ring_hom.id_apply",
"spectrum",
"weak_dual.character_space.exists_apply_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
spectrum.gelfand_transform_eq (a : A) : spectrum ℂ (gelfand_transform ℂ A a) = spectrum ℂ a | begin
ext z,
rw [continuous_map.spectrum_eq_range, weak_dual.character_space.mem_spectrum_iff_exists],
exact iff.rfl,
end | lemma | spectrum.gelfand_transform_eq | analysis.normed_space.star | src/analysis/normed_space/star/gelfand_duality.lean | [
"analysis.normed_space.star.spectrum",
"analysis.normed.group.quotient",
"analysis.normed_space.algebra",
"topology.continuous_function.units",
"topology.continuous_function.compact",
"topology.algebra.algebra",
"topology.continuous_function.stone_weierstrass"
] | [
"continuous_map.spectrum_eq_range",
"spectrum",
"weak_dual.character_space.mem_spectrum_iff_exists"
] | The Gelfand transform is spectrum-preserving. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gelfand_transform_map_star (a : A) :
gelfand_transform ℂ A (star a) = star (gelfand_transform ℂ A a) | continuous_map.ext $ λ φ, map_star φ a | lemma | gelfand_transform_map_star | analysis.normed_space.star | src/analysis/normed_space/star/gelfand_duality.lean | [
"analysis.normed_space.star.spectrum",
"analysis.normed.group.quotient",
"analysis.normed_space.algebra",
"topology.continuous_function.units",
"topology.continuous_function.compact",
"topology.algebra.algebra",
"topology.continuous_function.stone_weierstrass"
] | [
"continuous_map.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gelfand_transform_isometry : isometry (gelfand_transform ℂ A) | begin
nontriviality A,
refine add_monoid_hom_class.isometry_of_norm (gelfand_transform ℂ A) (λ a, _),
/- By `spectrum.gelfand_transform_eq`, the spectra of `star a * a` and its
`gelfand_transform` coincide. Therefore, so do their spectral radii, and since they are
self-adjoint, so also do their norms. Applyin... | lemma | gelfand_transform_isometry | analysis.normed_space.star | src/analysis/normed_space/star/gelfand_duality.lean | [
"analysis.normed_space.star.spectrum",
"analysis.normed.group.quotient",
"analysis.normed_space.algebra",
"topology.continuous_function.units",
"topology.continuous_function.compact",
"topology.algebra.algebra",
"topology.continuous_function.stone_weierstrass"
] | [
"cstar_ring.nnnorm_star_mul_self",
"ennreal.coe_eq_coe",
"gelfand_transform_map_star",
"is_self_adjoint.star_mul_self",
"isometry",
"map_mul",
"nnreal.sqrt_sq",
"spectral_radius",
"spectrum.gelfand_transform_eq"
] | The Gelfand transform is an isometry when the algebra is a C⋆-algebra over `ℂ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gelfand_transform_bijective : function.bijective (gelfand_transform ℂ A) | begin
refine ⟨(gelfand_transform_isometry A).injective, _⟩,
suffices : (gelfand_transform ℂ A).range = ⊤,
{ exact λ x, this.symm ▸ (gelfand_transform ℂ A).mem_range.mp (this.symm ▸ algebra.mem_top) },
/- Because the `gelfand_transform ℂ A` is an isometry, it has closed range, and so by the
Stone-Weierstrass t... | lemma | gelfand_transform_bijective | analysis.normed_space.star | src/analysis/normed_space/star/gelfand_duality.lean | [
"analysis.normed_space.star.spectrum",
"analysis.normed.group.quotient",
"analysis.normed_space.algebra",
"topology.continuous_function.units",
"topology.continuous_function.compact",
"topology.algebra.algebra",
"topology.continuous_function.stone_weierstrass"
] | [
"alg_hom.coe_to_ring_hom",
"alg_hom.to_ring_hom_eq_coe",
"algebra.mem_top",
"continuous_linear_map.ext",
"continuous_map.ext",
"continuous_map.subalgebra_is_R_or_C_topological_closure_eq_top_of_separates_points",
"gelfand_transform_isometry",
"gelfand_transform_map_star",
"le_rfl",
"subalgebra.le_... | The Gelfand transform is bijective when the algebra is a C⋆-algebra over `ℂ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gelfand_star_transform : A ≃⋆ₐ[ℂ] C(character_space ℂ A, ℂ) | star_alg_equiv.of_bijective
(show A →⋆ₐ[ℂ] C(character_space ℂ A, ℂ),
from { map_star' := λ x, gelfand_transform_map_star x, .. gelfand_transform ℂ A })
(gelfand_transform_bijective A) | def | gelfand_star_transform | analysis.normed_space.star | src/analysis/normed_space/star/gelfand_duality.lean | [
"analysis.normed_space.star.spectrum",
"analysis.normed.group.quotient",
"analysis.normed_space.algebra",
"topology.continuous_function.units",
"topology.continuous_function.compact",
"topology.algebra.algebra",
"topology.continuous_function.stone_weierstrass"
] | [
"gelfand_transform_bijective",
"gelfand_transform_map_star",
"star_alg_equiv.of_bijective"
] | The Gelfand transform as a `star_alg_equiv` between a commutative unital C⋆-algebra over `ℂ`
and the continuous functions on its `character_space`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_continuous_map (ψ : A →⋆ₐ[ℂ] B) :
C(character_space ℂ B, character_space ℂ A) | { to_fun := λ φ, equiv_alg_hom.symm ((equiv_alg_hom φ).comp (ψ.to_alg_hom)),
continuous_to_fun := continuous.subtype_mk (continuous_of_continuous_eval $
λ a, map_continuous $ gelfand_transform ℂ B (ψ a)) _ } | def | weak_dual.character_space.comp_continuous_map | analysis.normed_space.star | src/analysis/normed_space/star/gelfand_duality.lean | [
"analysis.normed_space.star.spectrum",
"analysis.normed.group.quotient",
"analysis.normed_space.algebra",
"topology.continuous_function.units",
"topology.continuous_function.compact",
"topology.algebra.algebra",
"topology.continuous_function.stone_weierstrass"
] | [
"continuous.subtype_mk"
] | The functorial map taking `ψ : A →⋆ₐ[ℂ] B` to a continuous function
`character_space ℂ B → character_space ℂ A` obtained by pre-composition with `ψ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_continuous_map_id :
comp_continuous_map (star_alg_hom.id ℂ A) = continuous_map.id (character_space ℂ A) | continuous_map.ext $ λ a, ext $ λ x, rfl | lemma | weak_dual.character_space.comp_continuous_map_id | analysis.normed_space.star | src/analysis/normed_space/star/gelfand_duality.lean | [
"analysis.normed_space.star.spectrum",
"analysis.normed.group.quotient",
"analysis.normed_space.algebra",
"topology.continuous_function.units",
"topology.continuous_function.compact",
"topology.algebra.algebra",
"topology.continuous_function.stone_weierstrass"
] | [
"continuous_map.ext",
"continuous_map.id",
"star_alg_hom.id"
] | `weak_dual.character_space.comp_continuous_map` sends the identity to the identity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_continuous_map_comp (ψ₂ : B →⋆ₐ[ℂ] C) (ψ₁ : A →⋆ₐ[ℂ] B) :
comp_continuous_map (ψ₂.comp ψ₁) = (comp_continuous_map ψ₁).comp (comp_continuous_map ψ₂) | continuous_map.ext $ λ a, ext $ λ x, rfl | lemma | weak_dual.character_space.comp_continuous_map_comp | analysis.normed_space.star | src/analysis/normed_space/star/gelfand_duality.lean | [
"analysis.normed_space.star.spectrum",
"analysis.normed.group.quotient",
"analysis.normed_space.algebra",
"topology.continuous_function.units",
"topology.continuous_function.compact",
"topology.algebra.algebra",
"topology.continuous_function.stone_weierstrass"
] | [
"continuous_map.ext"
] | `weak_dual.character_space.comp_continuous_map` is functorial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
entry_norm_bound_of_unitary {U : matrix n n 𝕜} (hU : U ∈ matrix.unitary_group n 𝕜) (i j : n):
‖U i j‖ ≤ 1 | begin
-- The norm squared of an entry is at most the L2 norm of its row.
have norm_sum : ‖ U i j ‖^2 ≤ (∑ x, ‖ U i x ‖^2),
{ apply multiset.single_le_sum,
{ intros x h_x,
rw multiset.mem_map at h_x,
cases h_x with a h_a,
rw ← h_a.2,
apply sq_nonneg },
{ rw multiset.mem_map,
u... | lemma | entry_norm_bound_of_unitary | analysis.normed_space.star | src/analysis/normed_space/star/matrix.lean | [
"analysis.matrix",
"analysis.normed_space.basic",
"data.is_R_or_C.basic",
"linear_algebra.unitary_group"
] | [
"finset.mem_univ_val",
"is_R_or_C.ext_iff",
"is_R_or_C.mul_conj",
"is_R_or_C.norm_sq_eq_def'",
"is_R_or_C.of_real_pow",
"is_R_or_C.one_re",
"matrix",
"matrix.conj_transpose_apply",
"matrix.mul_apply",
"matrix.one_apply_eq",
"matrix.unitary_group",
"mul_eq_one",
"multiset.mem_map",
"sq_eq_s... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
entrywise_sup_norm_bound_of_unitary {U : matrix n n 𝕜} (hU : U ∈ matrix.unitary_group n 𝕜) :
‖ U ‖ ≤ 1 | begin
simp_rw pi_norm_le_iff_of_nonneg zero_le_one,
intros i j,
exact entry_norm_bound_of_unitary hU _ _
end | lemma | entrywise_sup_norm_bound_of_unitary | analysis.normed_space.star | src/analysis/normed_space/star/matrix.lean | [
"analysis.matrix",
"analysis.normed_space.basic",
"data.is_R_or_C.basic",
"linear_algebra.unitary_group"
] | [
"entry_norm_bound_of_unitary",
"matrix",
"matrix.unitary_group",
"zero_le_one"
] | The entrywise sup norm of a unitary matrix is at most 1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_nnnorm_mul : ‖mul 𝕜 E a‖₊ = ‖a‖₊ | begin
rw ←Sup_closed_unit_ball_eq_nnnorm,
refine cSup_eq_of_forall_le_of_forall_lt_exists_gt _ _ (λ r hr, _),
{ exact (metric.nonempty_closed_ball.mpr zero_le_one).image _ },
{ rintro - ⟨x, hx, rfl⟩,
exact ((mul 𝕜 E a).unit_le_op_norm x $ mem_closed_ball_zero_iff.mp hx).trans
(op_norm_mul_apply_le 𝕜... | lemma | op_nnnorm_mul | analysis.normed_space.star | src/analysis/normed_space/star/mul.lean | [
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm"
] | [
"cSup_eq_of_forall_le_of_forall_lt_exists_gt",
"cstar_ring.nnnorm_self_mul_star",
"div_eq_mul_inv",
"inv_ne_zero",
"mul_inv",
"mul_lt_mul_of_pos_right",
"mul_smul_comm",
"nnnorm_smul",
"nnreal.le_inv_iff_mul_le",
"nnreal.lt_inv_iff_mul_lt",
"norm_smul",
"normed_field.exists_lt_nnnorm_lt",
"o... | In a C⋆-algebra `E`, either unital or non-unital, multiplication on the left by `a : E` has
norm equal to the norm of `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_nnnorm_mul_flip : ‖(mul 𝕜 E).flip a‖₊ = ‖a‖₊ | begin
rw [←Sup_unit_ball_eq_nnnorm, ←nnnorm_star, ←@op_nnnorm_mul 𝕜 E, ←Sup_unit_ball_eq_nnnorm],
congr' 1,
simp only [mul_apply', flip_apply],
refine set.subset.antisymm _ _;
rintro - ⟨b, hb, rfl⟩;
refine ⟨star b, by simpa only [norm_star, mem_ball_zero_iff] using hb, _⟩,
{ simp only [←star_mul, nnnorm_... | lemma | op_nnnorm_mul_flip | analysis.normed_space.star | src/analysis/normed_space/star/mul.lean | [
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm"
] | [
"nnnorm_star",
"op_nnnorm_mul",
"set.subset.antisymm"
] | In a C⋆-algebra `E`, either unital or non-unital, multiplication on the right by `a : E` has
norm eqaul to the norm of `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_isometry : isometry (mul 𝕜 E) | add_monoid_hom_class.isometry_of_norm _ (λ a, congr_arg coe $ op_nnnorm_mul 𝕜 a) | lemma | mul_isometry | analysis.normed_space.star | src/analysis/normed_space/star/mul.lean | [
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm"
] | [
"isometry",
"op_nnnorm_mul"
] | In a C⋆-algebra `E`, either unital or non-unital, the left regular representation is an
isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_flip_isometry : isometry (mul 𝕜 E).flip | add_monoid_hom_class.isometry_of_norm _ (λ a, congr_arg coe $ op_nnnorm_mul_flip 𝕜 a) | lemma | mul_flip_isometry | analysis.normed_space.star | src/analysis/normed_space/star/mul.lean | [
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm"
] | [
"isometry",
"op_nnnorm_mul_flip"
] | In a C⋆-algebra `E`, either unital or non-unital, the right regular anti-representation is an
isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
double_centralizer (𝕜 : Type u) (A : Type v) [nontrivially_normed_field 𝕜]
[non_unital_normed_ring A] [normed_space 𝕜 A] [smul_comm_class 𝕜 A A] [is_scalar_tower 𝕜 A A]
extends (A →L[𝕜] A) × (A →L[𝕜] A) | (central : ∀ x y : A, snd x * y = x * fst y) | structure | double_centralizer | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [
"is_scalar_tower",
"non_unital_normed_ring",
"nontrivially_normed_field",
"normed_space",
"smul_comm_class"
] | The type of *double centralizers*, also known as the *multiplier algebra* and denoted by
`𝓜(𝕜, A)`, of a non-unital normed algebra.
If `x : 𝓜(𝕜, A)`, then `x.fst` and `x.snd` are what is usually referred to as $L$ and $R$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
range_to_prod : set.range to_prod = {lr : (A →L[𝕜] A) × _ | ∀ x y, lr.2 x * y = x * lr.1 y} | set.ext $ λ x, ⟨by {rintro ⟨a, rfl⟩, exact a.central}, λ hx, ⟨⟨x, hx⟩, rfl⟩⟩ | lemma | double_centralizer.range_to_prod | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [
"set.ext",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_to_prod (s : S) (a : 𝓜(𝕜, A)) : (s • a).to_prod = s • a.to_prod | rfl | lemma | double_centralizer.smul_to_prod | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_fst (s : S) (a : 𝓜(𝕜, A)) : (s • a).fst = s • a.fst | rfl | lemma | double_centralizer.smul_fst | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_snd (s : S) (a : 𝓜(𝕜, A)) : (s • a).snd = s • a.snd | rfl | lemma | double_centralizer.smul_snd | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_to_prod (a b : 𝓜(𝕜, A)) : (a + b).to_prod = a.to_prod + b.to_prod | rfl | lemma | double_centralizer.add_to_prod | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_to_prod : (0 : 𝓜(𝕜, A)).to_prod = 0 | rfl | lemma | double_centralizer.zero_to_prod | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_to_prod (a : 𝓜(𝕜, A)) : (-a).to_prod = -a.to_prod | rfl | lemma | double_centralizer.neg_to_prod | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_to_prod (a b : 𝓜(𝕜, A)) : (a - b).to_prod = a.to_prod - b.to_prod | rfl | lemma | double_centralizer.sub_to_prod | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_to_prod : (1 : 𝓜(𝕜, A)).to_prod = 1 | rfl | lemma | double_centralizer.one_to_prod | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_to_prod (n : ℕ) : (n : 𝓜(𝕜 , A)).to_prod = n | rfl | lemma | double_centralizer.nat_cast_to_prod | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_cast_to_prod (n : ℤ) : (n : 𝓜(𝕜 , A)).to_prod = n | rfl | lemma | double_centralizer.int_cast_to_prod | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_to_prod (n : ℕ) (a : 𝓜(𝕜, A)) : (a ^ n).to_prod = a.to_prod ^ n | rfl | lemma | double_centralizer.pow_to_prod | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_fst (a b : 𝓜(𝕜, A)) : (a + b).fst = a.fst + b.fst | rfl | lemma | double_centralizer.add_fst | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_snd (a b : 𝓜(𝕜, A)) : (a + b).snd = a.snd + b.snd | rfl | lemma | double_centralizer.add_snd | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_fst : (0 : 𝓜(𝕜, A)).fst = 0 | rfl | lemma | double_centralizer.zero_fst | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_snd : (0 : 𝓜(𝕜, A)).snd = 0 | rfl | lemma | double_centralizer.zero_snd | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_fst (a : 𝓜(𝕜, A)) : (-a).fst = -a.fst | rfl | lemma | double_centralizer.neg_fst | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_snd (a : 𝓜(𝕜, A)) : (-a).snd = -a.snd | rfl | lemma | double_centralizer.neg_snd | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_fst (a b : 𝓜(𝕜, A)) : (a - b).fst = a.fst - b.fst | rfl | lemma | double_centralizer.sub_fst | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_snd (a b : 𝓜(𝕜, A)) : (a - b).snd = a.snd - b.snd | rfl | lemma | double_centralizer.sub_snd | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_fst : (1 : 𝓜(𝕜, A)).fst = 1 | rfl | lemma | double_centralizer.one_fst | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_snd : (1 : 𝓜(𝕜, A)).snd = 1 | rfl | lemma | double_centralizer.one_snd | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_fst (a b : 𝓜(𝕜, A)) : (a * b).fst = a.fst * b.fst | rfl | lemma | double_centralizer.mul_fst | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_snd (a b : 𝓜(𝕜, A)) : (a * b).snd = b.snd * a.snd | rfl | lemma | double_centralizer.mul_snd | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_fst (n : ℕ) : (n : 𝓜(𝕜 , A)).fst = n | rfl | lemma | double_centralizer.nat_cast_fst | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_snd (n : ℕ) : (n : 𝓜(𝕜 , A)).snd = n | rfl | lemma | double_centralizer.nat_cast_snd | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_cast_fst (n : ℤ) : (n : 𝓜(𝕜 , A)).fst = n | rfl | lemma | double_centralizer.int_cast_fst | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_cast_snd (n : ℤ) : (n : 𝓜(𝕜 , A)).snd = n | rfl | lemma | double_centralizer.int_cast_snd | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_fst (n : ℕ) (a : 𝓜(𝕜, A)) : (a ^ n).fst = a.fst ^ n | rfl | lemma | double_centralizer.pow_fst | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_snd (n : ℕ) (a : 𝓜(𝕜, A)) : (a ^ n).snd = a.snd ^ n | rfl | lemma | double_centralizer.pow_snd | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_prod_mul_opposite : 𝓜(𝕜, A) → (A →L[𝕜] A) × (A →L[𝕜] A)ᵐᵒᵖ | λ a, (a.fst, mul_opposite.op a.snd) | def | double_centralizer.to_prod_mul_opposite | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [
"mul_opposite.op"
] | The natural injection from `double_centralizer.to_prod` except the second coordinate inherits
`mul_opposite.op`. The ring structure on `𝓜(𝕜, A)` is the pullback under this map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_prod_mul_opposite_injective :
function.injective (to_prod_mul_opposite : 𝓜(𝕜, A) → (A →L[𝕜] A) × (A →L[𝕜] A)ᵐᵒᵖ) | λ a b h, let h' := prod.ext_iff.mp h in ext _ _ $ prod.ext h'.1 $ mul_opposite.op_injective h'.2 | lemma | double_centralizer.to_prod_mul_opposite_injective | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [
"mul_opposite.op_injective",
"prod.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_to_prod_mul_opposite :
set.range to_prod_mul_opposite = {lr : (A →L[𝕜] A) × _ | ∀ x y, unop lr.2 x * y = x * lr.1 y} | set.ext $ λ x,
⟨by {rintro ⟨a, rfl⟩, exact a.central}, λ hx, ⟨⟨(x.1, unop x.2), hx⟩, prod.ext rfl rfl⟩⟩ | lemma | double_centralizer.range_to_prod_mul_opposite | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [
"prod.ext",
"set.ext",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_prod_hom : 𝓜(𝕜, A) →+ (A →L[𝕜] A) × (A →L[𝕜] A) | { to_fun := to_prod,
map_zero' := rfl,
map_add' := λ x y, rfl } | def | double_centralizer.to_prod_hom | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | The canonical map `double_centralizer.to_prod` as an additive group homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_prod_mul_opposite_hom : 𝓜(𝕜, A) →+* (A →L[𝕜] A) × (A →L[𝕜] A)ᵐᵒᵖ | { to_fun := to_prod_mul_opposite,
map_zero' := rfl,
map_one' := rfl,
map_add' := λ x y, rfl,
map_mul' := λ x y, rfl } | def | double_centralizer.to_prod_mul_opposite_hom | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | The canonical map `double_centralizer.to_prod_mul_opposite` as a ring homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra_map_to_prod (k : 𝕜) :
(algebra_map 𝕜 𝓜(𝕜, A) k).to_prod = algebra_map 𝕜 _ k | rfl | lemma | double_centralizer.algebra_map_to_prod | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_fst (k : 𝕜) : (algebra_map 𝕜 𝓜(𝕜, A) k).fst = algebra_map 𝕜 _ k | rfl | lemma | double_centralizer.algebra_map_fst | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_snd (k : 𝕜) : (algebra_map 𝕜 𝓜(𝕜, A) k).snd = algebra_map 𝕜 _ k | rfl | lemma | double_centralizer.algebra_map_snd | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_fst (a : 𝓜(𝕜, A)) (b : A) : (star a).fst b = star (a.snd (star b)) | rfl | lemma | double_centralizer.star_fst | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_snd (a : 𝓜(𝕜, A)) (b : A) : (star a).snd b = star (a.fst (star b)) | rfl | lemma | double_centralizer.star_snd | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fst (a : A) : (a : 𝓜(𝕜, A)).fst = continuous_linear_map.mul 𝕜 A a | rfl | lemma | double_centralizer.coe_fst | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [
"continuous_linear_map.mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_snd (a : A) : (a : 𝓜(𝕜, A)).snd = (continuous_linear_map.mul 𝕜 A).flip a | rfl | lemma | double_centralizer.coe_snd | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [
"continuous_linear_map.mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq_algebra_map : (coe : 𝕜 → 𝓜(𝕜, 𝕜)) = algebra_map 𝕜 𝓜(𝕜, 𝕜) | begin
ext;
simp only [coe_fst, mul_apply', mul_one, algebra_map_to_prod, prod.algebra_map_apply, coe_snd,
flip_apply, one_mul];
simp only [algebra.algebra_map_eq_smul_one, smul_apply, one_apply, smul_eq_mul, mul_one],
end | lemma | double_centralizer.coe_eq_algebra_map | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [
"algebra.algebra_map_eq_smul_one",
"algebra_map",
"mul_one",
"one_mul",
"prod.algebra_map_apply",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_hom [star_ring 𝕜] [star_ring A] [star_module 𝕜 A] [normed_star_group A] :
A →⋆ₙₐ[𝕜] 𝓜(𝕜, A) | { to_fun := λ a, a,
map_smul' := λ k a, by ext; simp only [coe_fst, coe_snd, continuous_linear_map.map_smul,
smul_fst, smul_snd],
map_zero' := by ext; simp only [coe_fst, coe_snd, map_zero, zero_fst, zero_snd],
map_add' := λ a b, by ext; simp only [coe_fst, coe_snd, map_add, add_fst, add_snd],
map_mul' := λ... | def | double_centralizer.coe_hom | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [
"continuous_linear_map.coe_mul",
"continuous_linear_map.map_smul",
"mul_assoc",
"normed_star_group",
"star_module",
"star_ring",
"star_star"
] | The coercion of an algebra into its multiplier algebra as a non-unital star algebra
homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_def (a : 𝓜(𝕜, A)) : ‖a‖ = ‖a.to_prod_hom‖ | rfl | lemma | double_centralizer.norm_def | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_def (a : 𝓜(𝕜, A)) : ‖a‖₊ = ‖a.to_prod_hom‖₊ | rfl | lemma | double_centralizer.nnnorm_def | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_def' (a : 𝓜(𝕜, A)) : ‖a‖ = ‖a.to_prod_mul_opposite_hom‖ | rfl | lemma | double_centralizer.norm_def' | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_def' (a : 𝓜(𝕜, A)) : ‖a‖₊ = ‖a.to_prod_mul_opposite_hom‖₊ | rfl | lemma | double_centralizer.nnnorm_def' | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding_to_prod_mul_opposite :
uniform_embedding (@to_prod_mul_opposite 𝕜 A _ _ _ _ _) | uniform_embedding_comap to_prod_mul_opposite_injective | lemma | double_centralizer.uniform_embedding_to_prod_mul_opposite | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [
"uniform_embedding",
"uniform_embedding_comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_fst_eq_snd (a : 𝓜(𝕜, A)) : ‖a.fst‖ = ‖a.snd‖ | begin
-- a handy lemma for this proof
have h0 : ∀ f : A →L[𝕜] A, ∀ C : ℝ≥0, (∀ b : A, ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊) → ‖f‖₊ ≤ C,
{ intros f C h,
have h1 : ∀ b, C * ‖f b‖₊ * ‖b‖₊ ≤ C * ‖f‖₊ * ‖b‖₊ ^ 2,
{ intros b,
convert mul_le_mul_right' (mul_le_mul_left' (f.le_op_nnnorm b) C) (‖b‖₊) using 1,
... | lemma | double_centralizer.norm_fst_eq_snd | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [
"cstar_ring.nnnorm_self_mul_star",
"cstar_ring.nnnorm_star_mul_self",
"div_pow",
"mul_comm",
"mul_le_mul_left'",
"mul_le_mul_right'",
"mul_self_div_self",
"nnnorm_mul_le",
"nnnorm_star",
"ring"
] | For `a : 𝓜(𝕜, A)`, the norms of `a.fst` and `a.snd` coincide, and hence these
also coincide with `‖a‖` which is `max (‖a.fst‖) (‖a.snd‖)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnnorm_fst_eq_snd (a : 𝓜(𝕜, A)) : ‖a.fst‖₊ = ‖a.snd‖₊ | subtype.ext $ norm_fst_eq_snd a | lemma | double_centralizer.nnnorm_fst_eq_snd | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_fst (a : 𝓜(𝕜, A)) : ‖a.fst‖ = ‖a‖ | by simp only [norm_def, to_prod_hom_apply, prod.norm_def, norm_fst_eq_snd, max_eq_right,
eq_self_iff_true] | lemma | double_centralizer.norm_fst | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [
"prod.norm_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_snd (a : 𝓜(𝕜, A)) : ‖a.snd‖ = ‖a‖ | by rw [←norm_fst, norm_fst_eq_snd] | lemma | double_centralizer.norm_snd | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_fst (a : 𝓜(𝕜, A)) : ‖a.fst‖₊ = ‖a‖₊ | subtype.ext (norm_fst a) | lemma | double_centralizer.nnnorm_fst | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_snd (a : 𝓜(𝕜, A)) : ‖a.snd‖₊ = ‖a‖₊ | subtype.ext (norm_snd a) | lemma | double_centralizer.nnnorm_snd | analysis.normed_space.star | src/analysis/normed_space/star/multiplier.lean | [
"algebra.star.star_alg_hom",
"analysis.normed_space.star.basic",
"analysis.normed_space.operator_norm",
"analysis.special_functions.pow.nnreal",
"analysis.normed_space.star.mul"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unitary.spectrum_subset_circle (u : unitary E) :
spectrum 𝕜 (u : E) ⊆ metric.sphere 0 1 | begin
nontriviality E,
refine λ k hk, mem_sphere_zero_iff_norm.mpr (le_antisymm _ _),
{ simpa only [cstar_ring.norm_coe_unitary u] using norm_le_norm_of_mem hk },
{ rw ←unitary.coe_to_units_apply u at hk,
have hnk := ne_zero_of_mem_of_unit hk,
rw [←inv_inv (unitary.to_units u), ←spectrum.map_inv, set.me... | lemma | unitary.spectrum_subset_circle | analysis.normed_space.star | src/analysis/normed_space/star/spectrum.lean | [
"analysis.normed_space.star.basic",
"analysis.normed_space.spectrum",
"analysis.special_functions.exponential",
"algebra.star.star_alg_hom"
] | [
"cstar_ring.norm_coe_unitary",
"inv_le_of_inv_le",
"metric.sphere",
"norm_inv",
"set.mem_inv",
"spectrum",
"unitary",
"unitary.to_units"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
spectrum.subset_circle_of_unitary {u : E} (h : u ∈ unitary E) :
spectrum 𝕜 u ⊆ metric.sphere 0 1 | unitary.spectrum_subset_circle ⟨u, h⟩ | lemma | spectrum.subset_circle_of_unitary | analysis.normed_space.star | src/analysis/normed_space/star/spectrum.lean | [
"analysis.normed_space.star.basic",
"analysis.normed_space.spectrum",
"analysis.special_functions.exponential",
"algebra.star.star_alg_hom"
] | [
"metric.sphere",
"spectrum",
"unitary",
"unitary.spectrum_subset_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_self_adjoint.spectral_radius_eq_nnnorm {a : A}
(ha : is_self_adjoint a) :
spectral_radius ℂ a = ‖a‖₊ | begin
have hconst : tendsto (λ n : ℕ, (‖a‖₊ : ℝ≥0∞)) at_top _ := tendsto_const_nhds,
refine tendsto_nhds_unique _ hconst,
convert (spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectral_radius (a : A)).comp
(nat.tendsto_pow_at_top_at_top_of_one_lt one_lt_two),
refine funext (λ n, _),
rw [function.comp_ap... | lemma | is_self_adjoint.spectral_radius_eq_nnnorm | analysis.normed_space.star | src/analysis/normed_space/star/spectrum.lean | [
"analysis.normed_space.star.basic",
"analysis.normed_space.spectrum",
"analysis.special_functions.exponential",
"algebra.star.star_alg_hom"
] | [
"ennreal.coe_pow",
"is_self_adjoint",
"nat.tendsto_pow_at_top_at_top_of_one_lt",
"one_lt_two",
"spectral_radius",
"spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectral_radius",
"tendsto_const_nhds",
"tendsto_nhds_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_star_normal.spectral_radius_eq_nnnorm (a : A) [is_star_normal a] :
spectral_radius ℂ a = ‖a‖₊ | begin
refine (ennreal.pow_strict_mono two_ne_zero).injective _,
have heq : (λ n : ℕ, ((‖(a⋆ * a) ^ n‖₊ ^ (1 / n : ℝ)) : ℝ≥0∞))
= (λ x, x ^ 2) ∘ (λ n : ℕ, ((‖a ^ n‖₊ ^ (1 / n : ℝ)) : ℝ≥0∞)),
{ funext,
rw [function.comp_apply, ←rpow_nat_cast, ←rpow_mul, mul_comm, rpow_mul, rpow_nat_cast,
←coe_pow, sq,... | lemma | is_star_normal.spectral_radius_eq_nnnorm | analysis.normed_space.star | src/analysis/normed_space/star/spectrum.lean | [
"analysis.normed_space.star.basic",
"analysis.normed_space.spectrum",
"analysis.special_functions.exponential",
"algebra.star.star_alg_hom"
] | [
"commute.mul_pow",
"ennreal.continuous_pow",
"ennreal.pow_strict_mono",
"function.comp_apply",
"is_self_adjoint.star_mul_self",
"is_star_normal",
"mul_comm",
"spectral_radius",
"spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectral_radius",
"star_comm_self'",
"star_pow",
"tendsto_nhds_unique",... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_self_adjoint.mem_spectrum_eq_re [star_module ℂ A] {a : A}
(ha : is_self_adjoint a) {z : ℂ} (hz : z ∈ spectrum ℂ a) : z = z.re | begin
have hu := exp_mem_unitary_of_mem_skew_adjoint ℂ (ha.smul_mem_skew_adjoint conj_I),
let Iu := units.mk0 I I_ne_zero,
have : exp ℂ (I • z) ∈ spectrum ℂ (exp ℂ (I • a)),
by simpa only [units.smul_def, units.coe_mk0]
using spectrum.exp_mem_exp (Iu • a) (smul_mem_smul_iff.mpr hz),
exact complex.ext ... | theorem | is_self_adjoint.mem_spectrum_eq_re | analysis.normed_space.star | src/analysis/normed_space/star/spectrum.lean | [
"analysis.normed_space.star.basic",
"analysis.normed_space.spectrum",
"analysis.special_functions.exponential",
"algebra.star.star_alg_hom"
] | [
"complex.ext",
"exp",
"exp_mem_unitary_of_mem_skew_adjoint",
"is_self_adjoint",
"real.exp_eq_one_iff",
"smul_eq_mul",
"spectrum",
"spectrum.exp_mem_exp",
"spectrum.subset_circle_of_unitary",
"star_module",
"units.coe_mk0",
"units.mk0",
"units.smul_def"
] | Any element of the spectrum of a selfadjoint is real. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
self_adjoint.mem_spectrum_eq_re [star_module ℂ A]
(a : self_adjoint A) {z : ℂ} (hz : z ∈ spectrum ℂ (a : A)) : z = z.re | a.prop.mem_spectrum_eq_re hz | theorem | self_adjoint.mem_spectrum_eq_re | analysis.normed_space.star | src/analysis/normed_space/star/spectrum.lean | [
"analysis.normed_space.star.basic",
"analysis.normed_space.spectrum",
"analysis.special_functions.exponential",
"algebra.star.star_alg_hom"
] | [
"self_adjoint",
"spectrum",
"star_module"
] | Any element of the spectrum of a selfadjoint is real. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_self_adjoint.coe_re_map_spectrum [star_module ℂ A] {a : A}
(ha : is_self_adjoint a) : spectrum ℂ a = (coe ∘ re '' (spectrum ℂ a) : set ℂ) | le_antisymm (λ z hz, ⟨z, hz, (ha.mem_spectrum_eq_re hz).symm⟩) (λ z, by
{ rintros ⟨z, hz, rfl⟩,
simpa only [(ha.mem_spectrum_eq_re hz).symm, function.comp_app] using hz }) | theorem | is_self_adjoint.coe_re_map_spectrum | analysis.normed_space.star | src/analysis/normed_space/star/spectrum.lean | [
"analysis.normed_space.star.basic",
"analysis.normed_space.spectrum",
"analysis.special_functions.exponential",
"algebra.star.star_alg_hom"
] | [
"is_self_adjoint",
"spectrum",
"star_module"
] | The spectrum of a selfadjoint is real | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
self_adjoint.coe_re_map_spectrum [star_module ℂ A] (a : self_adjoint A) :
spectrum ℂ (a : A) = (coe ∘ re '' (spectrum ℂ (a : A)) : set ℂ) | a.property.coe_re_map_spectrum | theorem | self_adjoint.coe_re_map_spectrum | analysis.normed_space.star | src/analysis/normed_space/star/spectrum.lean | [
"analysis.normed_space.star.basic",
"analysis.normed_space.spectrum",
"analysis.special_functions.exponential",
"algebra.star.star_alg_hom"
] | [
"self_adjoint",
"spectrum",
"star_module"
] | The spectrum of a selfadjoint is real | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnnorm_apply_le (a : A) : ‖(φ a : B)‖₊ ≤ ‖a‖₊ | begin
suffices : ∀ s : A, is_self_adjoint s → ‖φ s‖₊ ≤ ‖s‖₊,
{ exact nonneg_le_nonneg_of_sq_le_sq zero_le'
(by simpa only [nnnorm_star_mul_self, map_star, map_mul]
using this _ (is_self_adjoint.star_mul_self a)) },
{ intros s hs,
simpa only [hs.spectral_radius_eq_nnnorm, (hs.star_hom_apply φ).spec... | lemma | star_alg_hom.nnnorm_apply_le | analysis.normed_space.star | src/analysis/normed_space/star/spectrum.lean | [
"analysis.normed_space.star.basic",
"analysis.normed_space.spectrum",
"analysis.special_functions.exponential",
"algebra.star.star_alg_hom"
] | [
"alg_hom.spectrum_apply_subset",
"is_self_adjoint",
"is_self_adjoint.star_mul_self",
"map_mul",
"nonneg_le_nonneg_of_sq_le_sq",
"spectral_radius",
"supr_le_supr_of_subset",
"zero_le'"
] | A star algebra homomorphism of complex C⋆-algebras is norm contractive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_apply_le (a : A) : ‖(φ a : B)‖ ≤ ‖a‖ | nnnorm_apply_le φ a | lemma | star_alg_hom.norm_apply_le | analysis.normed_space.star | src/analysis/normed_space/star/spectrum.lean | [
"analysis.normed_space.star.basic",
"analysis.normed_space.spectrum",
"analysis.special_functions.exponential",
"algebra.star.star_alg_hom"
] | [] | A star algebra homomorphism of complex C⋆-algebras is norm contractive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.alg_hom_class.star_alg_hom_class : star_alg_hom_class F ℂ A ℂ | { coe := λ f, f,
.. weak_dual.complex.star_hom_class,
.. hF } | def | alg_hom_class.star_alg_hom_class | analysis.normed_space.star | src/analysis/normed_space/star/spectrum.lean | [
"analysis.normed_space.star.basic",
"analysis.normed_space.spectrum",
"analysis.special_functions.exponential",
"algebra.star.star_alg_hom"
] | [
"star_alg_hom_class"
] | This is not an instance to avoid type class inference loops. See
`weak_dual.complex.star_hom_class`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gronwall_bound (δ K ε x : ℝ) : ℝ | if K = 0 then δ + ε * x else δ * exp (K * x) + (ε / K) * (exp (K * x) - 1) | def | gronwall_bound | analysis.ODE | src/analysis/ODE/gronwall.lean | [
"analysis.special_functions.exp_deriv"
] | [
"exp"
] | Upper bound used in several Grönwall-like inequalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gronwall_bound_K0 (δ ε : ℝ) : gronwall_bound δ 0 ε = λ x, δ + ε * x | funext $ λ x, if_pos rfl | lemma | gronwall_bound_K0 | analysis.ODE | src/analysis/ODE/gronwall.lean | [
"analysis.special_functions.exp_deriv"
] | [
"gronwall_bound"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gronwall_bound_of_K_ne_0 {δ K ε : ℝ} (hK : K ≠ 0) :
gronwall_bound δ K ε = λ x, δ * exp (K * x) + (ε / K) * (exp (K * x) - 1) | funext $ λ x, if_neg hK | lemma | gronwall_bound_of_K_ne_0 | analysis.ODE | src/analysis/ODE/gronwall.lean | [
"analysis.special_functions.exp_deriv"
] | [
"exp",
"gronwall_bound"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_gronwall_bound (δ K ε x : ℝ) :
has_deriv_at (gronwall_bound δ K ε) (K * (gronwall_bound δ K ε x) + ε) x | begin
by_cases hK : K = 0,
{ subst K,
simp only [gronwall_bound_K0, zero_mul, zero_add],
convert ((has_deriv_at_id x).const_mul ε).const_add δ,
rw [mul_one] },
{ simp only [gronwall_bound_of_K_ne_0 hK],
convert (((has_deriv_at_id x).const_mul K).exp.const_mul δ).add
((((has_deriv_at_id x).co... | lemma | has_deriv_at_gronwall_bound | analysis.ODE | src/analysis/ODE/gronwall.lean | [
"analysis.special_functions.exp_deriv"
] | [
"gronwall_bound",
"gronwall_bound_K0",
"gronwall_bound_of_K_ne_0",
"has_deriv_at",
"has_deriv_at_id",
"mul_assoc",
"mul_comm",
"mul_div_cancel'",
"mul_one",
"ring",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_gronwall_bound_shift (δ K ε x a : ℝ) :
has_deriv_at (λ y, gronwall_bound δ K ε (y - a)) (K * (gronwall_bound δ K ε (x - a)) + ε) x | begin
convert (has_deriv_at_gronwall_bound δ K ε _).comp x ((has_deriv_at_id x).sub_const a),
rw [id, mul_one]
end | lemma | has_deriv_at_gronwall_bound_shift | analysis.ODE | src/analysis/ODE/gronwall.lean | [
"analysis.special_functions.exp_deriv"
] | [
"gronwall_bound",
"has_deriv_at",
"has_deriv_at_gronwall_bound",
"has_deriv_at_id",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gronwall_bound_x0 (δ K ε : ℝ) : gronwall_bound δ K ε 0 = δ | begin
by_cases hK : K = 0,
{ simp only [gronwall_bound, if_pos hK, mul_zero, add_zero] },
{ simp only [gronwall_bound, if_neg hK, mul_zero, exp_zero, sub_self, mul_one, add_zero] }
end | lemma | gronwall_bound_x0 | analysis.ODE | src/analysis/ODE/gronwall.lean | [
"analysis.special_functions.exp_deriv"
] | [
"exp_zero",
"gronwall_bound",
"mul_one",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gronwall_bound_ε0 (δ K x : ℝ) : gronwall_bound δ K 0 x = δ * exp (K * x) | begin
by_cases hK : K = 0,
{ simp only [gronwall_bound_K0, hK, zero_mul, exp_zero, add_zero, mul_one] },
{ simp only [gronwall_bound_of_K_ne_0 hK, zero_div, zero_mul, add_zero] }
end | lemma | gronwall_bound_ε0 | analysis.ODE | src/analysis/ODE/gronwall.lean | [
"analysis.special_functions.exp_deriv"
] | [
"exp",
"exp_zero",
"gronwall_bound",
"gronwall_bound_K0",
"gronwall_bound_of_K_ne_0",
"mul_one",
"zero_div",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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