Lemma 7.4.7 (Characters).label Let $\frak{m}\tleq A$ be a maximal regular ideal of an abelian Banach algebra. The quotient algebra $A/\frak{m}$ is a field which admits a unique (algebra) isomorphism to $\com$. Consequently, there exists a unique surjective (algebra) homomorphism by $\om_{\frak{m}}:A\to \com$ whose kernel is $\frak{m}$
Proof. Since $\frak{m}$ is maximal regular ideal, Proposition 7.4.3 and Proposition 7.4.5 imply the quotient algebra $A/\frak{m}$ is a unital Banach algebra with no nontrivial proper ideal. Hence by Proposition 7.4.6, it follows that $A/\frak{m}$ is a field. Therefore by Theorem 7.3.8, $\exists\p:A/\frak{m}\iso \com$ and if $\psi:A/\frak{m}\iso \com$ then $\psi\circ\p\inv$ is a linear automorphism of $\com$ hence the identity thus $\p$ is unique. We extend this to a surjective homomorphism
where $\pi$ is the quotient map $A\to A/\frak{m}$. Since $\p$ is an isomorphism $\om_{\frak{m}}\inv(0)=\pi\inv(0)=\frak{m}$$\square$
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