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Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Instances of the continuous functional calculus #

Main theorems #

IsSelfAdjoint.instContinuousFunctionalCalculus: the continuous functional calculus for selfadjoint elements in a ℂ-algebra with a continuous functional calculus for normal elements and where every element has compact spectrum. In particular, this includes unital C⋆-algebras over ℂ.
Nonneg.instContinuousFunctionalCalculus: the continuous functional calculus for nonnegative elements in an ℝ-algebra with a continuous functional calculus for selfadjoint elements, where every element has compact spectrum, and where nonnegative elements have nonnegative spectrum. In particular, this includes unital C⋆-algebras over ℝ.

Tags #

continuous functional calculus, normal, selfadjoint

Pull back a non-unital instance from a unital one on the unitization #

noncomputable def cfcₙAux {𝕜 : Type u_1} {A : Type u_2} [RCLike 𝕜] [NonUnitalNormedRing A] [StarRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [StarModule 𝕜 A] {p : A → Prop} {p₁ : Unitization 𝕜 A → Prop} (hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x) (a : A) (ha : p a) [ContinuousFunctionalCalculus 𝕜 p₁] :

ContinuousMapZero (↑(quasispectrum 𝕜 a)) 𝕜 →⋆ₙₐ[𝕜] Unitization 𝕜 A

This is an auxiliary definition used for constructing an instance of the non-unital continuous functional calculus given a instance of the unital one on the unitization.

This is the natural non-unital star homomorphism obtained from the chain

calc
  C(σₙ 𝕜 a, 𝕜)₀ →⋆ₙₐ[𝕜] C(σₙ 𝕜 a, 𝕜) := ContinuousMapZero.toContinuousMapHom
  _             ≃⋆[𝕜] C(σ 𝕜 (↑a : A⁺¹), 𝕜) := Homeomorph.compStarAlgEquiv'
  _             →⋆ₐ[𝕜] A⁺¹ := cfcHom

This range of this map is contained in the range of (↑) : A → A⁺¹ (see cfcₙAux_mem_range_inr), and so we may restrict it to A to get the necessary homomorphism for the non-unital continuous functional calculus.

Equations

cfcₙAux hp₁ a ha = ((↑(cfcHom ⋯)).comp ↑(Homeomorph.compStarAlgEquiv' 𝕜 𝕜 (Homeomorph.setCongr ⋯))).comp ContinuousMapZero.toContinuousMapHom

Instances For

theorem cfcₙAux_id {𝕜 : Type u_1} {A : Type u_2} [RCLike 𝕜] [NonUnitalNormedRing A] [StarRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [StarModule 𝕜 A] {p : A → Prop} {p₁ : Unitization 𝕜 A → Prop} (hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x) (a : A) (ha : p a) [ContinuousFunctionalCalculus 𝕜 p₁] :

(cfcₙAux ⋯ a ha) (ContinuousMapZero.id ⋯) = ↑a

theorem isClosedEmbedding_cfcₙAux {𝕜 : Type u_1} {A : Type u_2} [RCLike 𝕜] [NonUnitalNormedRing A] [StarRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [StarModule 𝕜 A] {p : A → Prop} {p₁ : Unitization 𝕜 A → Prop} (hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x) (a : A) (ha : p a) [ContinuousFunctionalCalculus 𝕜 p₁] :

Topology.IsClosedEmbedding ⇑(cfcₙAux ⋯ a ha)

@[deprecated isClosedEmbedding_cfcₙAux (since := "2024-10-20")]

theorem closedEmbedding_cfcₙAux {𝕜 : Type u_1} {A : Type u_2} [RCLike 𝕜] [NonUnitalNormedRing A] [StarRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [StarModule 𝕜 A] {p : A → Prop} {p₁ : Unitization 𝕜 A → Prop} (hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x) (a : A) (ha : p a) [ContinuousFunctionalCalculus 𝕜 p₁] :

Topology.IsClosedEmbedding ⇑(cfcₙAux ⋯ a ha)

Alias of isClosedEmbedding_cfcₙAux.

theorem spec_cfcₙAux {𝕜 : Type u_1} {A : Type u_2} [RCLike 𝕜] [NonUnitalNormedRing A] [StarRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [StarModule 𝕜 A] {p : A → Prop} {p₁ : Unitization 𝕜 A → Prop} (hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x) (a : A) (ha : p a) [ContinuousFunctionalCalculus 𝕜 p₁] (f : ContinuousMapZero (↑(quasispectrum 𝕜 a)) 𝕜) :

spectrum 𝕜 ((cfcₙAux ⋯ a ha) f) = Set.range ⇑f

theorem cfcₙAux_mem_range_inr {𝕜 : Type u_1} {A : Type u_2} [RCLike 𝕜] [NonUnitalNormedRing A] [StarRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [StarModule 𝕜 A] {p : A → Prop} {p₁ : Unitization 𝕜 A → Prop} (hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x) (a : A) (ha : p a) [ContinuousFunctionalCalculus 𝕜 p₁] [CompleteSpace A] (f : ContinuousMapZero (↑(quasispectrum 𝕜 a)) 𝕜) :

(cfcₙAux ⋯ a ha) f ∈ NonUnitalStarAlgHom.range (Unitization.inrNonUnitalStarAlgHom 𝕜 A)

theorem RCLike.nonUnitalContinuousFunctionalCalculus {𝕜 : Type u_1} {A : Type u_2} [RCLike 𝕜] [NonUnitalNormedRing A] [StarRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [StarModule 𝕜 A] {p : A → Prop} {p₁ : Unitization 𝕜 A → Prop} (hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x) [ContinuousFunctionalCalculus 𝕜 p₁] [CompleteSpace A] [CStarRing A] :

NonUnitalContinuousFunctionalCalculus 𝕜 p

Continuous functional calculus for selfadjoint elements #

theorem isSelfAdjoint_iff_isStarNormal_and_quasispectrumRestricts {A : Type u_1} [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [Module ℂ A] [IsScalarTower ℂ A A] [SMulCommClass ℂ A A] [NonUnitalContinuousFunctionalCalculus ℂ IsStarNormal] {a : A} :

IsSelfAdjoint a ↔ IsStarNormal a ∧ QuasispectrumRestricts a ⇑Complex.reCLM

An element in a non-unital C⋆-algebra is selfadjoint if and only if it is normal and its quasispectrum is contained in ℝ.

theorem IsSelfAdjoint.quasispectrumRestricts {A : Type u_1} [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [Module ℂ A] [IsScalarTower ℂ A A] [SMulCommClass ℂ A A] [NonUnitalContinuousFunctionalCalculus ℂ IsStarNormal] {a : A} :

IsSelfAdjoint a → IsStarNormal a ∧ QuasispectrumRestricts a ⇑Complex.reCLM

Alias of the forward direction of isSelfAdjoint_iff_isStarNormal_and_quasispectrumRestricts.

An element in a non-unital C⋆-algebra is selfadjoint if and only if it is normal and its quasispectrum is contained in ℝ.

theorem QuasispectrumRestricts.isSelfAdjoint {A : Type u_1} [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [Module ℂ A] [IsScalarTower ℂ A A] [SMulCommClass ℂ A A] [NonUnitalContinuousFunctionalCalculus ℂ IsStarNormal] (a : A) (ha : QuasispectrumRestricts a ⇑Complex.reCLM) [IsStarNormal a] :

IsSelfAdjoint a

A normal element whose ℂ-quasispectrum is contained in ℝ is selfadjoint.

instance IsSelfAdjoint.instNonUnitalContinuousFunctionalCalculus {A : Type u_1} [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [Module ℂ A] [IsScalarTower ℂ A A] [SMulCommClass ℂ A A] [NonUnitalContinuousFunctionalCalculus ℂ IsStarNormal] :

NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint

theorem isSelfAdjoint_iff_isStarNormal_and_spectrumRestricts {A : Type u_1} [TopologicalSpace A] [Ring A] [StarRing A] [Algebra ℂ A] [ContinuousFunctionalCalculus ℂ IsStarNormal] {a : A} :

IsSelfAdjoint a ↔ IsStarNormal a ∧ SpectrumRestricts a ⇑Complex.reCLM

An element in a C⋆-algebra is selfadjoint if and only if it is normal and its spectrum is contained in ℝ.

theorem IsSelfAdjoint.spectrumRestricts {A : Type u_1} [TopologicalSpace A] [Ring A] [StarRing A] [Algebra ℂ A] [ContinuousFunctionalCalculus ℂ IsStarNormal] {a : A} (ha : IsSelfAdjoint a) :

SpectrumRestricts a ⇑Complex.reCLM

theorem SpectrumRestricts.isSelfAdjoint {A : Type u_1} [TopologicalSpace A] [Ring A] [StarRing A] [Algebra ℂ A] [ContinuousFunctionalCalculus ℂ IsStarNormal] (a : A) (ha : SpectrumRestricts a ⇑Complex.reCLM) [IsStarNormal a] :

IsSelfAdjoint a

A normal element whose ℂ-spectrum is contained in ℝ is selfadjoint.

instance IsSelfAdjoint.instContinuousFunctionalCalculus {A : Type u_1} [TopologicalSpace A] [Ring A] [StarRing A] [Algebra ℂ A] [ContinuousFunctionalCalculus ℂ IsStarNormal] :

ContinuousFunctionalCalculus ℝ IsSelfAdjoint

theorem IsSelfAdjoint.spectrum_nonempty {A : Type u_2} [Ring A] [StarRing A] [TopologicalSpace A] [Algebra ℝ A] [ContinuousFunctionalCalculus ℝ IsSelfAdjoint] [Nontrivial A] {a : A} (ha : IsSelfAdjoint a) :

(spectrum ℝ a).Nonempty

Continuous functional calculus for nonnegative elements #

theorem CFC.exists_sqrt_of_isSelfAdjoint_of_quasispectrumRestricts {A : Type u_1} [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module ℝ A] [IsScalarTower ℝ A A] [SMulCommClass ℝ A A] [NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint] {a : A} (ha₁ : IsSelfAdjoint a) (ha₂ : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal) :

∃ (x : A), IsSelfAdjoint x ∧ QuasispectrumRestricts x ⇑ContinuousMap.realToNNReal ∧ x * x = a

theorem nonneg_iff_isSelfAdjoint_and_quasispectrumRestricts {A : Type u_1} [NonUnitalRing A] [PartialOrder A] [StarRing A] [StarOrderedRing A] [TopologicalSpace A] [Module ℝ A] [IsScalarTower ℝ A A] [SMulCommClass ℝ A A] [NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint] [NonnegSpectrumClass ℝ A] {a : A} :

0 ≤ a ↔ IsSelfAdjoint a ∧ QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal

instance Nonneg.instNonUnitalContinuousFunctionalCalculus {A : Type u_1} [NonUnitalRing A] [PartialOrder A] [StarRing A] [StarOrderedRing A] [TopologicalSpace A] [Module ℝ A] [IsScalarTower ℝ A A] [SMulCommClass ℝ A A] [NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint] [NonnegSpectrumClass ℝ A] :

NonUnitalContinuousFunctionalCalculus NNReal fun (x : A) => 0 ≤ x

theorem NNReal.spectrum_nonempty {A : Type u_2} [Ring A] [StarRing A] [PartialOrder A] [TopologicalSpace A] [Algebra NNReal A] [ContinuousFunctionalCalculus NNReal fun (x : A) => 0 ≤ x] [Nontrivial A] {a : A} (ha : 0 ≤ a) :

(spectrum NNReal a).Nonempty

theorem CFC.exists_sqrt_of_isSelfAdjoint_of_spectrumRestricts {A : Type u_1} [Ring A] [StarRing A] [TopologicalSpace A] [Algebra ℝ A] [ContinuousFunctionalCalculus ℝ IsSelfAdjoint] {a : A} (ha₁ : IsSelfAdjoint a) (ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal) :

∃ (x : A), IsSelfAdjoint x ∧ SpectrumRestricts x ⇑ContinuousMap.realToNNReal ∧ x ^ 2 = a

theorem nonneg_iff_isSelfAdjoint_and_spectrumRestricts {A : Type u_1} [Ring A] [PartialOrder A] [StarRing A] [StarOrderedRing A] [TopologicalSpace A] [Algebra ℝ A] [ContinuousFunctionalCalculus ℝ IsSelfAdjoint] [NonnegSpectrumClass ℝ A] {a : A} :

0 ≤ a ↔ IsSelfAdjoint a ∧ SpectrumRestricts a ⇑ContinuousMap.realToNNReal

instance Nonneg.instContinuousFunctionalCalculus {A : Type u_1} [Ring A] [PartialOrder A] [StarRing A] [StarOrderedRing A] [TopologicalSpace A] [Algebra ℝ A] [ContinuousFunctionalCalculus ℝ IsSelfAdjoint] [NonnegSpectrumClass ℝ A] :

ContinuousFunctionalCalculus NNReal fun (x : A) => 0 ≤ x

The restriction of a continuous functional calculus is equal to the original one #

theorem cfcHom_real_eq_restrict {A : Type u_1} [TopologicalSpace A] [Ring A] [StarRing A] [Algebra ℂ A] [ContinuousFunctionalCalculus ℂ IsStarNormal] [T2Space A] {a : A} (ha : IsSelfAdjoint a) :

cfcHom ha = SpectrumRestricts.starAlgHom (cfcHom ⋯) ⋯

theorem cfc_real_eq_complex {A : Type u_1} [TopologicalSpace A] [Ring A] [StarRing A] [Algebra ℂ A] [ContinuousFunctionalCalculus ℂ IsStarNormal] [T2Space A] {a : A} (f : ℝ → ℝ) (ha : IsSelfAdjoint a := by cfc_tac) :

cfc f a = cfc (fun (x : ℂ) => ↑(f x.re)) a

theorem cfcₙHom_real_eq_restrict {A : Type u_1} [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [Module ℂ A] [IsScalarTower ℂ A A] [SMulCommClass ℂ A A] [T2Space A] [NonUnitalContinuousFunctionalCalculus ℂ IsStarNormal] {a : A} (ha : IsSelfAdjoint a) :

cfcₙHom ha = QuasispectrumRestricts.nonUnitalStarAlgHom (cfcₙHom ⋯) ⋯

theorem cfcₙ_real_eq_complex {A : Type u_1} [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [Module ℂ A] [IsScalarTower ℂ A A] [SMulCommClass ℂ A A] [T2Space A] [NonUnitalContinuousFunctionalCalculus ℂ IsStarNormal] {a : A} (f : ℝ → ℝ) (ha : IsSelfAdjoint a := by cfc_tac) :

cfcₙ f a = cfcₙ (fun (x : ℂ) => ↑(f x.re)) a

theorem cfcHom_nnreal_eq_restrict {A : Type u_1} [TopologicalSpace A] [Ring A] [PartialOrder A] [StarRing A] [StarOrderedRing A] [Algebra ℝ A] [TopologicalRing A] [T2Space A] [ContinuousFunctionalCalculus ℝ IsSelfAdjoint] [NonnegSpectrumClass ℝ A] {a : A} (ha : 0 ≤ a) :

cfcHom ha = SpectrumRestricts.starAlgHom (cfcHom ⋯) ⋯

theorem cfc_nnreal_eq_real {A : Type u_1} [TopologicalSpace A] [Ring A] [PartialOrder A] [StarRing A] [StarOrderedRing A] [Algebra ℝ A] [TopologicalRing A] [T2Space A] [ContinuousFunctionalCalculus ℝ IsSelfAdjoint] [NonnegSpectrumClass ℝ A] {a : A} (f : NNReal → NNReal) (ha : 0 ≤ a := by cfc_tac) :

cfc f a = cfc (fun (x : ℝ) => ↑(f x.toNNReal)) a

theorem cfcₙHom_nnreal_eq_restrict {A : Type u_1} [TopologicalSpace A] [NonUnitalRing A] [PartialOrder A] [StarRing A] [StarOrderedRing A] [Module ℝ A] [TopologicalRing A] [IsScalarTower ℝ A A] [SMulCommClass ℝ A A] [T2Space A] [NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint] [NonnegSpectrumClass ℝ A] {a : A} (ha : 0 ≤ a) :

cfcₙHom ha = QuasispectrumRestricts.nonUnitalStarAlgHom (cfcₙHom ⋯) ⋯

theorem cfcₙ_nnreal_eq_real {A : Type u_1} [TopologicalSpace A] [NonUnitalRing A] [PartialOrder A] [StarRing A] [StarOrderedRing A] [Module ℝ A] [TopologicalRing A] [IsScalarTower ℝ A A] [SMulCommClass ℝ A A] [T2Space A] [NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint] [NonnegSpectrumClass ℝ A] {a : A} (f : NNReal → NNReal) (ha : 0 ≤ a := by cfc_tac) :

cfcₙ f a = cfcₙ (fun (x : ℝ) => ↑(f x.toNNReal)) a