Corollary 34.7.6.label Let $A, B$ be unital $C^{*}$-algebras and $\Phi: A \to B$ be an injective unital *-homomorphism, then for each $x \in A$,
- (1)
$\sigma_{B}(\Phi(x)) = \sigma_{A}(x)$.
- (2)
$\norm{\Phi(x)}_{B} = \norm{x}_{A}$.
Proof, [II.10.7, Zhu93]. (1): Since $\Phi(G(A)) \subset G(B)$, $\sigma_{B}(\Phi(x)) \subset \sigma_{A}(x)$. If $\sigma_{B}(\Phi(x)) \subsetneq \sigma_{A}(x)$, then Urysohn’s Lemma implies that there exists $C(\sigma_{A}(x); \complex)$ such that $f|_{\sigma_B(\Phi(x))}= 0$ but $f \ne 0$. In which case, by (7) of the continuous functional calculus, $\Phi(f(x)) = f(\Phi(x)) = 0$, which contradicts the fact that $\Phi$ is injective.
(2): By Corollary 34.4.5, $\Phi$ is isometric.$\square$
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