Character space of an algebra (${\Phi }_A$)

The character space ${\Phi }_A$ is the set of all characters on $A$.

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.

For an algebra of functions on a space, evaluation at $x$ is the character ${\delta }_x(f)=f(x)$.

For a commutative Banach algebra $A$, the Gelfand transform sends $a\in A$ to the continuous function $\hat{a}(\phi )=\phi (a)$ on its character space of an algebra. It is a contractive unital algebra homomorphism into $C({\Phi }_A)$.