# 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`

.

The result we have established here is the strongest possible, but it is not the version which is
most useful in practice. The generic API for the continuous functional calculus can be found in
`Analysis.CStarAlgebra.ContinuousFunctionalCalculus`

in the `Unital`

and `NonUnital`

files. The
relevant instances on C⋆-algebra can be found in the `Instances`

file.

## 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.

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, ⋯⟩, ⋯⟩

## Instances For

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

## Instances For

**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.