---
tag: 9EHVQ8Q0A8
kind: definition
name: Continuous functional calculus
msc: [46L05, 47A60, 46H30]
defines: JB7S6WPZ77
uses: [EH0CZ19RKP, NT564Y315M, 0X3WGX95FE]
sources:
  - url: https://en.wikipedia.org/wiki/Continuous_functional_calculus
---

## Definition

Let $A$ be a C*-algebra with unit $1$, and let $a \in A$ be normal, that is, $a^* a = a a^*$.
Let $\sigma(a) \subseteq \C$ be the [[spectrum|EH0CZ19RKP]] of $a$, and let $C(\sigma(a))$ be the C\*-algebra of [[continuous|0X3WGX95FE]] functions $\sigma(a) \to \C$ with the sup norm.
There is a unique $*$-homomorphism $\Phi_a \colon C(\sigma(a)) \to A$ with $\Phi_a(1) = 1$ and $\Phi_a(\iota) = a$, where $\iota(z) = z$ is the inclusion of $\sigma(a)$ into $\C$.
The map $\Phi_a$ is the *continuous functional calculus* of $a$, and for $f \in C(\sigma(a))$ one writes $f(a) = \Phi_a(f)$.

## Remarks

Existence comes from the [[commutative Gelfand–Naimark theorem|NT564Y315M]] applied to the commutative C\*-algebra $C^*(a, 1)$ generated by $a$ and $1$, whose character space is homeomorphic to $\sigma(a)$.
Uniqueness holds because the polynomials in $z$ and $\overline{z}$ are dense in $C(\sigma(a))$, by the Stone–Weierstrass theorem.
The map $\Phi_a$ is isometric, its image is $C^*(a, 1)$, and $\sigma(f(a)) = f(\sigma(a))$.

Example: $A = M_n(\C)$ and $a = U D U^*$ with $U$ unitary and $D = \operatorname{diag}(\lambda_1, \ldots, \lambda_n)$.
Here $\sigma(a) = \set{\lambda_1, \ldots, \lambda_n}$ and $f(a) = U \operatorname{diag}(f(\lambda_1), \ldots, f(\lambda_n)) U^*$.
This map is linear, multiplicative, preserves adjoints, sends $1$ to $I$ and $\iota$ to $a$.

Non-example: $a = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ in $M_2(\C)$ is not normal.
Its spectrum is $\set{0}$, so $\iota$ is the zero function on $\sigma(a)$, and any linear map sends it to $0 \ne a$.
So no such $\Phi_a$ exists, and normality cannot be dropped.