Proposition 34.5.2.label Let $A, B$ be unital $C^{*}$-algebras and $\phi: A \to B$ be a unital *-homomorphism, then for each $x \in A$,
- (1)
$\sigma_{B}(\phi(x)) \subset \sigma_{A}(x)$.
- (2)
$\norm{\phi(x)}_{B} \le \norm{x}_{A}$.
Proof. (1): Since $\phi$ is unital, $\phi(G(A)) \subset G(B)$, so $\sigma_{B}(\phi(x)) \subset \sigma_{A}(x)$.
(2): By (1) and Corollary 34.3.5,
\begin{align*}\norm{\phi(x)}_{B}^{2}&= \sup\bracsn{|\lambda|\ | \lambda \in \sigma_B(\phi(x^*x))}\\&\ge \sup\bracsn{|\lambda|\ | \lambda \in \sigma_A(x^*x)}= \norm{x}_{A}^{2}\end{align*}
$\square$
Post a Comment