PlanetMath: $C^*$-algebra homomorphisms preserve continuous functional calculus

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-algebra homomorphisms preserve continuous functional calculus

(Theorem)

Let us setup some notation first: Let $\mathcal{A}$ be a unital -algebra and a normal element of $\mathcal{A}$ . Then

$\sigma(z)$ denotes the spectrum of .
$C(\sigma(z))$ denotes the -algebra of continuous functions $\sigma(z) \longrightarrow \mathbb{C}$ .
If $f \in C(\sigma(z))$ then is the element of $\mathcal{A}$ given by the continuous functional calculus.

$\,$

Theorem - Let $\mathcal{A}$ , $\mathcal{B}$ be unital -algebras and $\Phi :\mathcal{A} \longrightarrow \mathcal{B}$ a *-homomorphism. Let be a normal element in $\mathcal{A}$ . If $f \in C(\sigma(x))$ then

$\displaystyle \Phi(f(x)) = f(\Phi(x))$

$\,$

Proof: The identity elements of $\mathcal{A}$ and $\mathcal{B}$ will be both denoted by and it will be clear from the context which one we are referring to.

First, we need to check that $f(\Phi(x))$ is a well-defined element of $\mathcal{B}$ , i.e. that $\sigma(\Phi(x)) \subseteq \sigma(x)$ . This is clear since, if $x - \lambda e$ is invertible for some $\lambda \in \mathbb{C}$ , then $\Phi(x)-\lambda e = \Phi(x- \lambda e)$ is also invertible.

Let $\{p_n\}$ be sequence of polynomials in $C(\sigma(x))$ converging uniformly to . Then we have that

$\Phi(p_n(x)) \longrightarrow \Phi(f(x))$ , by the continuity of $\Phi$ (see this entry) and the continuity of the continuous functional calculus mapping.

$p_n(\Phi(x)) \longrightarrow f(\Phi(x))$ , by the continuity of the continuous functional calculus mapping.

It is easily checked that $\Phi(p_n(x)) = p_n(\Phi(x))$ (since $\Phi$ is an homomorphism). Hence we conclude that $\Phi(f(x)) = f(\Phi(x))$ as intended. $\square$

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Cross-references: homomorphism, mapping, polynomials, sequence, invertible, well-defined, clear, identity elements, proof, *-homomorphism, continuous functional calculus, continuous functions, spectrum, normal element, unital
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This is version 2 of $C^*$

-algebra homomorphisms preserve continuous functional calculus, born on 2008-04-21, modified 2008-04-21.
Object id is 10529, canonical name is CAlgebraHomomorphismsPreserveContinuousFunctionalCalculus.
Accessed 186 times total.

Classification:

AMS MSC:	46L05 (Functional analysis :: Selfadjoint operator algebras :: General theory of $C^*$-algebras)
	47A60 (Operator theory :: General theory of linear operators :: Functional calculus)

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