Character space

Remarks

Other names are the maximal ideal space and the Gelfand spectrum, written $\sigma(A)$ or $\mathfrak{M}(A)$. The space $\Delta(A)$ is locally compact and Hausdorff. If $A$ has a unit, then $\varphi(1) = 1$ for every character, $\Delta(A)$ is weak-* closed in the unit ball of $A^*$, and so it is compact by the Banach–Alaoglu theorem. For unital $A$, the map $\varphi \mapsto \ker \varphi$ is a bijection from $\Delta(A)$ onto the set of maximal ideals of $A$; the Gelfand–Mazur theorem shows that each quotient $A / \mathfrak{m}$ is $\C$.

Example: let $X$ be a compact Hausdorff space and $A = C(X)$ with the sup norm. For $x \in X$, the evaluation $\varphi_x(f) = f(x)$ is linear and multiplicative, and $\varphi_x(1) = 1$, so it is a character. Every character of $C(X)$ has this form, and $x \mapsto \varphi_x$ is a homeomorphism from $X$ onto $\Delta(A)$.

Non-example: on $C([0,1])$, the map $\varphi(f) = \int_0^1 f(t) \, dt$ is linear and nonzero but not multiplicative. For $f(t) = t$ one has $\varphi(f^2) = 1/3$, while $\varphi(f)^2 = 1/4$.