Proposition 33.7.3.label Let $A$ be a Banach algebra and $\phi \in A^{*}$ be a multiplicative functional, then $\norm{\phi}_{A^*}\le 1$.
Proof. Let $\tilde A$ be the unitisation of $A$, then by (U) of the unitisation, $\phi$ extends to a multiplicative functional $\tilde \phi$ on $\tilde A$. Therefore $\norm{\phi}_{A^*}\le \normn{\tilde \phi}_{{\tilde A}^*}= 1$.$\square$
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