${C}^{*}$algebra homomorphisms preserve continuous functional calculus
Let us setup some notation first: Let $\mathcal{A}$ be a unital ${C}^{*}$algebra^{} (http://planetmath.org/CAlgebra) and $z$ a normal element of $\mathcal{A}$. Then

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$\sigma (z)$ denotes the spectrum of $z$.

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$C(\sigma (z))$ denotes the ${C}^{*}$algebra of continuous functions^{} $\sigma (z)\u27f6\u2102$.

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If $f\in C(\sigma (z))$ then $f(z)$ is the element of $\mathcal{A}$ given by the continuous functional calculus.
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Theorem^{}  Let $\mathcal{A}$, $\mathcal{B}$ be unital ${C}^{*}$algebras (http://planetmath.org/CAlgebra) and $\mathrm{\Phi}:\mathcal{A}\u27f6\mathcal{B}$ a *homomorphism^{}. Let $x$ be a normal element in $\mathcal{A}$. If $f\in C(\sigma (x))$ then
$\mathrm{\Phi}(f(x))=f(\mathrm{\Phi}(x))$ 
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Proof: The identity elements^{} of $\mathcal{A}$ and $\mathcal{B}$ will be both denoted by $e$ and it will be clear from the context which one we are referring to.
First, we need to check that $f(\mathrm{\Phi}(x))$ is a welldefined element of $\mathcal{B}$, i.e. that $\sigma (\mathrm{\Phi}(x))\subseteq \sigma (x)$. This is clear since, if $x\lambda e$ is invertible^{} for some $\lambda \in \u2102$, then $\mathrm{\Phi}(x)\lambda e=\mathrm{\Phi}(x\lambda e)$ is also invertible.
Let $\{{p}_{n}\}$ be sequence of polynomials in $C(\sigma (x))$ converging uniformly to $f$. Then we have that

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$\mathrm{\Phi}({p}_{n}(x))\u27f6\mathrm{\Phi}(f(x))$, by the continuity of $\mathrm{\Phi}$ (see this entry (http://planetmath.org/HomomorphismsOfCAlgebrasAreContinuous)) and the continuity of the continuous functional calculus mapping.

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${p}_{n}(\mathrm{\Phi}(x))\u27f6f(\mathrm{\Phi}(x))$, by the continuity of the continuous functional calculus mapping.
It is easily checked that $\mathrm{\Phi}({p}_{n}(x))={p}_{n}(\mathrm{\Phi}(x))$ (since $\mathrm{\Phi}$ is an homomorphism). Hence we conclude that $\mathrm{\Phi}(f(x))=f(\mathrm{\Phi}(x))$ as intended. $\mathrm{\square}$
Title  ${C}^{*}$algebra homomorphisms preserve continuous functional calculus 

Canonical name  CalgebraHomomorphismsPreserveContinuousFunctionalCalculus 
Date of creation  20130322 18:00:50 
Last modified on  20130322 18:00:50 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  5 
Author  asteroid (17536) 
Entry type  Theorem 
Classification  msc 47A60 
Classification  msc 46L05 