Kernel of Character on Unital Commutative Banach Algebra is Maximal Ideal/Proof 1

Theorem

Let $\struct {A, \norm {\, \cdot \,} }$ be a unital commutative Banach algebra over $\C$.

Let $\phi : A \to \C$ be a character on $A$.

Then $\ker \phi$ is a maximal ideal of $A$.

Proof

From Kernel of Ring Homomorphism is Ideal, $\ker \phi$ is an ideal of $A$.

From the First Ring Isomorphism Theorem, we have:

$\phi \sqbrk A$ and $\dfrac A {\ker \phi}$ are isomorphic as rings.

From Character on Banach Algebra is Surjective, we have that $\phi \sqbrk A = \C$.

Hence:

$\dfrac A {\ker \phi} \cong \C$

That is:

$\dfrac A {\ker \phi}$ is a field.

From Maximal Ideal iff Quotient Ring is Field, we conclude that $\ker \phi$ is a maximal ideal of $A$.

$\blacksquare$