Character space

Summary

The character space of a commutative complex Banach algebra $A$ is the set of nonzero algebra homomorphisms $A \to \C$, with the weak-* topology. It is also called the maximal ideal space or the spectrum of $A$. For unital $A$, the characters correspond to the maximal ideals of $A$. It is a locally compact Hausdorff space, compact when $A$ has a unit. It is the domain of the Gelfand transform, which turns elements of $A$ into continuous functions.