Proposition 29.7.2.label Let $A$ be a unital Banach algebra and $\phi \in A^{*}$ be a multiplicative functional, then
- (1)
For each $x \in A$, $|\phi(x)| \le [x]_{sp}\le \norm{x}_{A}$.
- (2)
$\norm{\phi}_{A^*}= 1$.
- (3)
$\phi(G(A)) \subset \complex \setminus \bracs{0}$.
Proof. (3): For each $x \in G(A)$, $1 = \phi(xx^{-1}) = \phi(x)\phi(x^{-1}) \ne 0$.
(1): By (3), for every $\lambda \in \complex$ with $|\lambda| > [x]_{sp}$, $\lambda x \in G(A)$ and $\lambda - \phi(x) \ne 0$. Therefore $\phi(x) \in \ol{B(0, [x]_{sp})}$.
(2): For each $\lambda \in \complex$, $\phi(\lambda 1) = \lambda$, so $\norm{\phi}_{A^*}\le 1$.$\square$
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