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leanprover-community-bot committed Jun 11, 2023
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20 changes: 10 additions & 10 deletions Mathbin/Analysis/NormedSpace/Star/GelfandDuality.lean
Original file line number Diff line number Diff line change
Expand Up @@ -96,39 +96,39 @@ theorem Ideal.toCharacterSpace_apply_eq_zero_of_mem {a : A} (ha : a ∈ I) :

/-- 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. -/
theorem WeakDual.characterSpace.exists_apply_eq_zero {a : A} (ha : ¬IsUnit a) :
theorem WeakDual.CharacterSpace.exists_apply_eq_zero {a : A} (ha : ¬IsUnit a) :
∃ f : characterSpace ℂ A, f a = 0 :=
by
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⟩))⟩
#align weak_dual.character_space.exists_apply_eq_zero WeakDual.characterSpace.exists_apply_eq_zero
#align weak_dual.character_space.exists_apply_eq_zero WeakDual.CharacterSpace.exists_apply_eq_zero

theorem WeakDual.characterSpace.mem_spectrum_iff_exists {a : A} {z : ℂ} :
theorem WeakDual.CharacterSpace.mem_spectrum_iff_exists {a : A} {z : ℂ} :
z ∈ spectrum ℂ a ↔ ∃ f : characterSpace ℂ A, f a = z :=
by
refine' ⟨fun hz => _, _⟩
· obtain ⟨f, hf⟩ := WeakDual.characterSpace.exists_apply_eq_zero hz
· obtain ⟨f, hf⟩ := WeakDual.CharacterSpace.exists_apply_eq_zero hz
simp only [map_sub, sub_eq_zero, AlgHomClass.commutes, Algebra.id.map_eq_id,
RingHom.id_apply] at hf
exact (ContinuousMap.spectrum_eq_range (gelfand_transform ℂ A a)).symm ▸ ⟨f, hf.symm⟩
· rintro ⟨f, rfl⟩
exact AlgHom.apply_mem_spectrum f a
#align weak_dual.character_space.mem_spectrum_iff_exists WeakDual.characterSpace.mem_spectrum_iff_exists
#align weak_dual.character_space.mem_spectrum_iff_exists WeakDual.CharacterSpace.mem_spectrum_iff_exists

/-- The Gelfand transform is spectrum-preserving. -/
theorem spectrum.gelfandTransform_eq (a : A) : spectrum ℂ (gelfandTransform ℂ A a) = spectrum ℂ a :=
by
ext z
rw [ContinuousMap.spectrum_eq_range, WeakDual.characterSpace.mem_spectrum_iff_exists]
rw [ContinuousMap.spectrum_eq_range, WeakDual.CharacterSpace.mem_spectrum_iff_exists]
exact Iff.rfl
#align spectrum.gelfand_transform_eq spectrum.gelfandTransform_eq

instance [Nontrivial A] : Nonempty (characterSpace ℂ A) :=
⟨Classical.choose <|
WeakDual.characterSpace.exists_apply_eq_zero <| zero_mem_nonunits.2 zero_ne_one⟩
WeakDual.CharacterSpace.exists_apply_eq_zero <| zero_mem_nonunits.2 zero_ne_one⟩

end ComplexBanachAlgebra

Expand Down Expand Up @@ -230,7 +230,7 @@ noncomputable def compContinuousMap (ψ : A →⋆ₐ[ℂ] B) : C(characterSpace
continuous_toFun :=
Continuous.subtype_mk
(continuous_of_continuous_eval fun a => map_continuous <| gelfandTransform ℂ B (ψ a)) _
#align weak_dual.character_space.comp_continuous_map WeakDual.characterSpace.compContinuousMap
#align weak_dual.character_space.comp_continuous_map WeakDual.CharacterSpace.compContinuousMap

variable (A)

Expand All @@ -239,7 +239,7 @@ variable (A)
theorem compContinuousMap_id :
compContinuousMap (StarAlgHom.id ℂ A) = ContinuousMap.id (characterSpace ℂ A) :=
ContinuousMap.ext fun a => ext fun x => rfl
#align weak_dual.character_space.comp_continuous_map_id WeakDual.characterSpace.compContinuousMap_id
#align weak_dual.character_space.comp_continuous_map_id WeakDual.CharacterSpace.compContinuousMap_id

variable {A}

Expand All @@ -248,7 +248,7 @@ variable {A}
theorem compContinuousMap_comp (ψ₂ : B →⋆ₐ[ℂ] C) (ψ₁ : A →⋆ₐ[ℂ] B) :
compContinuousMap (ψ₂.comp ψ₁) = (compContinuousMap ψ₁).comp (compContinuousMap ψ₂) :=
ContinuousMap.ext fun a => ext fun x => rfl
#align weak_dual.character_space.comp_continuous_map_comp WeakDual.characterSpace.compContinuousMap_comp
#align weak_dual.character_space.comp_continuous_map_comp WeakDual.CharacterSpace.compContinuousMap_comp

end CharacterSpace

Expand Down
8 changes: 4 additions & 4 deletions lake-manifest.json
Original file line number Diff line number Diff line change
Expand Up @@ -10,15 +10,15 @@
{"git":
{"url": "https://github.com/leanprover-community/lean3port.git",
"subDir?": null,
"rev": "89e80ca6d8a5abbd5d629909990c85861edf3c51",
"rev": "f2e94385444a2c7dd316d2dda4f9c3b11dd3bd5b",
"name": "lean3port",
"inputRev?": "89e80ca6d8a5abbd5d629909990c85861edf3c51"}},
"inputRev?": "f2e94385444a2c7dd316d2dda4f9c3b11dd3bd5b"}},
{"git":
{"url": "https://github.com/leanprover-community/mathlib4.git",
"subDir?": null,
"rev": "fe6b4e5dec87688f16a85e35885110b315661925",
"rev": "95092b27655f26da45329eb20ab2d8773e8d4642",
"name": "mathlib",
"inputRev?": "fe6b4e5dec87688f16a85e35885110b315661925"}},
"inputRev?": "95092b27655f26da45329eb20ab2d8773e8d4642"}},
{"git":
{"url": "https://github.com/gebner/quote4",
"subDir?": null,
Expand Down
4 changes: 2 additions & 2 deletions lakefile.lean
Original file line number Diff line number Diff line change
Expand Up @@ -4,7 +4,7 @@ open Lake DSL System
-- Usually the `tag` will be of the form `nightly-2021-11-22`.
-- If you would like to use an artifact from a PR build,
-- it will be of the form `pr-branchname-sha`.
def tag : String := "nightly-2023-06-11-21"
def tag : String := "nightly-2023-06-11-23"
def releaseRepo : String := "leanprover-community/mathport"
def oleanTarName : String := "mathlib3-binport.tar.gz"

Expand Down Expand Up @@ -38,7 +38,7 @@ target fetchOleans (_pkg : Package) : Unit := do
untarReleaseArtifact releaseRepo tag oleanTarName libDir
return .nil

require lean3port from git "https://github.com/leanprover-community/lean3port.git"@"89e80ca6d8a5abbd5d629909990c85861edf3c51"
require lean3port from git "https://github.com/leanprover-community/lean3port.git"@"f2e94385444a2c7dd316d2dda4f9c3b11dd3bd5b"

@[default_target]
lean_lib Mathbin where
Expand Down

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