Proposition 29.8.2.label Let $A$ be a commutative unital Banach algebra and $x \in A$, then:
- (1)
$\Gamma_{A}$ is a contractive homomorphism.
- (2)
$\Gamma_{A}(1) = 1$.
- (3)
$x \in G(A)$ if and only if $\Gamma_{A} x \in G(C(\Omega(A); \complex))$.
- (4)
$(\Gamma_{A}x)(\Omega(A)) = \sigma_{A}(x)$.
- (5)
$\norm{\Gamma_Ax}_{u} = [x]_{sp}$.
Proof, [Theorem 1.1.13, Fol16]. (2): For each $\phi \in \Omega(A)$, $\phi(1) = 1$, so $\Gamma_{A}(1) = 1$.
(3): Since $A$ is commutative, $x \not\in G(A)$ if and only if the ideal generated by $x$ is proper, if and only if there exists a maximal ideal containing $x$, if and only if there exists $\phi \in \Omega(A)$ with $\phi(x) = 0$.
(4): By (1) and (3),
\[(\Gamma_{A}x)(\Omega(A)) = \sigma_{C(\Omega(A); \complex)}(\Gamma x) = \sigma_{A}(x)\]
$\square$
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