Proposition 29.7.4.label Let $A$ be a commutative unital Banach algebra, then the mapping
\[\Omega(A) \to \cm(A) \quad \phi \mapsto \ker(\phi)\]
is a bijection.
Proof. For each $\phi \in \cm(A)$, $\ker(\phi)$ is an ideal of codimension $1$, and must be maximal.
On the other hand, for each $I \in \Omega(A)$, the quotient $A/I$ is a field, so by the Gelfand-Mazur Theorem, it is isomorphic to $\complex$. Thus the canonical projection $A \to A/I \iso \complex$ induces a multiplicative functional on $A$.$\square$
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