Maximal Ideal in Unital Banach Algebra is Closed

Theorem

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

Let $I$ be a maximal ideal of $A$.

Then $I$ is closed.

Proof

From Closure of Proper Ideal in Unital Banach Algebra is Proper Ideal, the closure $I^-$ is a proper ideal of $A$ with $I \subseteq I^-$.

Since $I$ is a maximal ideal, we have $I = I^-$.

From Set is Closed iff Equals Topological Closure, we conclude that $I$ is closed.

$\blacksquare$