Character space of a commutative C*-algebra

Let $A$ be a commutative $C^*$ -algebra. Its character space, denoted $\Delta(A)$ or $\widehat A$ , is the set of all characters $\chi:A\to\mathbb C$ , equipped with the weak-star topology inherited from $A^*$ . Equivalently, it is the weakest topology making every evaluation map $\chi\mapsto\chi(a)$ continuous for $a\in A$ . The space $\Delta(A)$ is locally compact and Hausdorff. For nonzero $A$ , it is compact exactly when $A$ is unital. No topology is chosen independently of the algebra. This construction is intrinsic.

Topological structure

All characters have norm one. For a nonunital algebra, adjoining the zero functional compactifies $\Delta(A)$ inside the weak-star compact dual unit ball; this compactification agrees with the character space of the unitization, with the new character as the point at infinity. These facts explain why local compactness, rather than compactness, is the natural general setting.

Function-algebra model

If $A=C_0(X)$ for a locally compact Hausdorff space $X$ , the map

x\longmapsto\chi_x,\qquad \chi_x(f)=f(x),

is a homeomorphism $X\cong\Delta(A)$ . Under this identification, the topology reconstructed from evaluation of algebra elements is exactly the original topology of $X$ .

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: the chapter on characters, the Gelfand topology, and commutative $C^*$ -algebras.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: the introductory treatment of commutative $C^*$ -algebras and their spectra.