Proposition 29.8.3.label Let $A$ be a commutative unital Banach algebra, then the following are equivalent:
- (1)
For each $x \in A$, $\normn{x^2}_{A} = \norm{x}_{A}^{2}$.
- (2)
$\Gamma_{A}$ is an isometry.
Proof. (1) $\Rightarrow$ (2): For each $x \in A$, by the spectral radius formula and (5) of Proposition 29.8.2,
\[\norm{\Gamma_A x}_{u} = [x]_{sp}= \norm{x}_{A}\]
(2) $\Rightarrow$ (1): For each $x \in A$, by (5) of Proposition 29.8.2,
\[\normn{x^2}_{A} \ge [x^{2}]_{sp}= \normn{\Gamma_A x^2}_{u} = \normn{\Gamma_A x}_{u}^{2} = \normn{x}_{A}^{2}\]
$\square$
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