---
tag: M7JVFJ3MMX
kind: concept
name: Character space
msc: [46J05, 46J20]
sources:
  - url: https://en.wikipedia.org/wiki/Gelfand_representation
---

## 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|0PV74TR5FP]] of $A$.
It is a locally compact Hausdorff space, compact when $A$ has a unit.
It is the domain of the [[Gelfand transform|VWNSHKJT92]], which turns elements of $A$ into continuous functions.

## Explanations

### L1 human

A character is a way to turn each element of the algebra into a single complex number while respecting sums and products.
For the algebra of continuous functions on a compact space $X$, the characters are exactly the evaluations $f \mapsto f(x)$ at points $x \in X$.
So the character space of $C(X)$ is $X$ itself.
For a general commutative Banach algebra, the character space is the space of "points" on which its elements become functions.