Continuous functional calculus

In this file we construct the continuousFunctionalCalculus for a normal element a of a (unital) C⋆-algebra over ℂ. This is a star algebra equivalence C(spectrum ℂ a, ℂ) ≃⋆ₐ[ℂ] elementalStarAlgebra ℂ a which sends the (restriction of) the identity map ContinuousMap.id ℂ to the (unique) preimage of a under the coercion of elementalStarAlgebra ℂ a to A.

Being a star algebra equivalence between C⋆-algebras, this map is continuous (even an isometry), and by the Stone-Weierstrass theorem it is the unique star algebra equivalence which extends the polynomial functional calculus (i.e., Polynomial.aeval).

For any continuous function f : spectrum ℂ a → ℂ, this makes it possible to define an element f a (not valid notation) in the original algebra, which heuristically has the same eigenspaces as a and acts on eigenvector of a for an eigenvalue λ as multiplication by f λ. This description is perfectly accurate in finite dimension, but only heuristic in infinite dimension as there might be no genuine eigenvector. In particular, when f is a polynomial ∑ cᵢ Xⁱ, then f a is ∑ cᵢ aⁱ. Also, id a = a.

This file also includes a proof of the spectral permanence theorem for (unital) C⋆-algebras (see StarSubalgebra.spectrum_eq)

Main definitions

• continuousFunctionalCalculus : C(spectrum ℂ a, ℂ) ≃⋆ₐ[ℂ] elementalStarAlgebra ℂ a: this is the composition of the inverse of the gelfandStarTransform with the natural isomorphism induced by the homeomorphism elementalStarAlgebra.characterSpaceHomeo.
• elementalStarAlgebra.characterSpaceHomeo : characterSpace ℂ (elementalStarAlgebra ℂ a) ≃ₜ spectrum ℂ a: this homeomorphism is defined by evaluating a character φ at a, and noting that φ a ∈ spectrum ℂ a since φ is an algebra homomorphism. Moreover, this map is continuous and bijective and since the spaces involved are compact Hausdorff, it is a homeomorphism.

Main statements

• StarSubalgebra.coe_isUnit: for x : S in a C⋆-Subalgebra S of A, then ↑x : A is a Unit if and only if x is a unit.
• StarSubalgebra.spectrum_eq: spectral_permanence for x : S, where S is a C⋆-Subalgebra of A, spectrum ℂ x = spectrum ℂ (x : A).

Notes

The result we have established here is the strongest possible, but it is likely not the version which will be most useful in practice. Future work will include developing an appropriate API for the continuous functional calculus (including one for real-valued functions with real argument that applies to self-adjoint elements of the algebra).

instance instNormedCommRingSubtypeMemStarSubalgebraElementalStarAlgebraOfIsStarNormal {R : Type u_2} {A : Type u_3} [CommRing R] [StarRing R] [NormedRing A] [Algebra R A] [StarRing A] [ContinuousStar A] [StarModule R A] (a : A) [IsStarNormal a] : NormedCommRing ↥(elementalStarAlgebra R a)

Equations

• instNormedCommRingSubtypeMemStarSubalgebraElementalStarAlgebraOfIsStarNormal a = let __src := SubringClass.toNormedRing (elementalStarAlgebra R a); NormedCommRing.mk ⋯

theorem spectrum_star_mul_self_of_isStarNormal {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] (a : A) [IsStarNormal a] : spectrum ℂ (star a * a) ⊆ Set.Icc 0 ↑‖star a * a‖

This lemma is used in the proof of elementalStarAlgebra.isUnit_of_isUnit_of_isStarNormal, which in turn is the key to spectral permanence StarSubalgebra.spectrum_eq, which is itself necessary for the continuous functional calculus. Using the continuous functional calculus, this lemma can be superseded by one that omits the IsStarNormal hypothesis.

theorem elementalStarAlgebra.isUnit_of_isUnit_of_isStarNormal {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] {a : A} [IsStarNormal a] (h : IsUnit a) : IsUnit ⟨a, ⋯⟩

This is the key lemma on the way to establishing spectral permanence for C⋆-algebras, which is established in StarSubalgebra.spectrum_eq. This lemma is superseded by StarSubalgebra.coe_isUnit, which does not require an IsStarNormal hypothesis and holds for any closed star subalgebra.

theorem StarSubalgebra.isUnit_coe_inv_mem {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] {S : StarSubalgebra ℂ A} (hS : IsClosed ↑S) {x : A} (h : IsUnit x) (hxS : x ∈ S) : ↑h.unit⁻¹ ∈ S

For x : A which is invertible in A, the inverse lies in any unital C⋆-subalgebra S containing x.

theorem StarSubalgebra.coe_isUnit {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] {S : StarSubalgebra ℂ A} (hS : IsClosed ↑S) {x : ↥S} : IsUnit ↑x ↔ IsUnit x

For a unital C⋆-subalgebra S of A and x : S, if ↑x : A is invertible in A, then x is invertible in S.

theorem StarSubalgebra.mem_spectrum_iff {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] {S : StarSubalgebra ℂ A} (hS : IsClosed ↑S) {x : ↥S} {z : ℂ} : z ∈ spectrum ℂ x ↔ z ∈ spectrum ℂ ↑x

theorem StarSubalgebra.spectrum_eq {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] {S : StarSubalgebra ℂ A} (hS : IsClosed ↑S) (x : ↥S) : spectrum ℂ x = spectrum ℂ ↑x

Spectral permanence. The spectrum of an element is invariant of the (closed) StarSubalgebra in which it is contained.

@[simp]

theorem elementalStarAlgebra.characterSpaceToSpectrum_coe {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] (x : A) (φ : ↑(WeakDual.characterSpace ℂ ↥(elementalStarAlgebra ℂ x))) : ↑(elementalStarAlgebra.characterSpaceToSpectrum x φ) = φ ⟨x, ⋯⟩

noncomputable def elementalStarAlgebra.characterSpaceToSpectrum {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] (x : A) (φ : ↑(WeakDual.characterSpace ℂ ↥(elementalStarAlgebra ℂ x))) : ↑(spectrum ℂ x)

The natural map from characterSpace ℂ (elementalStarAlgebra ℂ x) to spectrum ℂ x given by evaluating φ at x. This is essentially just evaluation of the gelfandTransform of x, but because we want something in spectrum ℂ x, as opposed to spectrum ℂ ⟨x, elementalStarAlgebra.self_mem ℂ x⟩ there is slightly more work to do.

Equations

• elementalStarAlgebra.characterSpaceToSpectrum x φ = ⟨φ ⟨x, ⋯⟩, ⋯⟩

theorem elementalStarAlgebra.continuous_characterSpaceToSpectrum {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] (x : A) : Continuous (elementalStarAlgebra.characterSpaceToSpectrum x)

theorem elementalStarAlgebra.bijective_characterSpaceToSpectrum {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] (a : A) [IsStarNormal a] : Function.Bijective (elementalStarAlgebra.characterSpaceToSpectrum a)

noncomputable def elementalStarAlgebra.characterSpaceHomeo {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] (a : A) [IsStarNormal a] : ↑(WeakDual.characterSpace ℂ ↥(elementalStarAlgebra ℂ a)) ≃ₜ ↑(spectrum ℂ a)

The homeomorphism between the character space of the unital C⋆-subalgebra generated by a single normal element a : A and spectrum ℂ a.

Equations

• elementalStarAlgebra.characterSpaceHomeo a = ⋯.homeoOfEquivCompactToT2

noncomputable def continuousFunctionalCalculus {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] (a : A) [IsStarNormal a] : C(↑(spectrum ℂ a), ℂ) ≃⋆ₐ[ℂ] ↥(elementalStarAlgebra ℂ a)

Continuous functional calculus. Given a normal element a : A of a unital C⋆-algebra, the continuous functional calculus is a StarAlgEquiv from the complex-valued continuous functions on the spectrum of a to the unital C⋆-subalgebra generated by a. Moreover, this equivalence identifies (ContinuousMap.id ℂ).restrict (spectrum ℂ a)) with a; see continuousFunctionalCalculus_map_id. As such it extends the polynomial functional calculus.

Equations

• continuousFunctionalCalculus a = (Homeomorph.compStarAlgEquiv' ℂ ℂ (elementalStarAlgebra.characterSpaceHomeo a)).trans (gelfandStarTransform ↥(elementalStarAlgebra ℂ a)).symm

theorem continuousFunctionalCalculus_map_id {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [CstarRing A] [StarModule ℂ A] [CompleteSpace A] (a : A) [IsStarNormal a] : (continuousFunctionalCalculus a) (ContinuousMap.restrict (spectrum ℂ a) (ContinuousMap.id ℂ)) = ⟨a, ⋯⟩