Definition 32.4.3 (Maximal Ideal).label Let $A$ be a Banach algebra and $I \subset A$ be an ideal, then $I$ is maximal if:

1. (1)

   $I$ is proper.

2. (2)

   For any proper ideal $J \subsetneq A$ with $J \supset I$, $I = J$.

The set $\cm(A)$ of maximal two-sided ideals of $A$ is the maximal ideal space of $A$.