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A character on a complex algebra is a nonzero multiplicative linear functional $\phi :A\to \mathbb{C}$.

Algebra characters form the domain of the Gelfand representation; they are distinct from traces of group representations.

Every commutative unital C-star algebra $A$ is isometrically star-isomorphic to $C({\Phi }_A)$ through its Gelfand transform, where ${\Phi }_A$ is its character space of an algebra. Thus abstract algebra elements can be treated as continuous functions on a compact Hausdorff space.

This is the isometric commutative C-star specialization of the Gelfand representation.

The Gelfand topology on ${\Phi }_A$ is the weak-star topology inherited from $A^{\ast }$, equivalently the coarsest topology making every map $\phi \mapsto \phi (a)$ continuous.

This is the topology on the character space used in the Gelfand representation.