Example 1

Let X be a locally compact topological space and consider the commutative Banach algebra $\mathscr{C}_0(X)$ (example 3 of I, p. 17 and No. 2 of I, p. 31). By Corollary 1 of I, p. 32, the character space $\mathsf{X}(\mathscr{C}_0(X))$ is identified with X by means of the mapping which to an element $x\in X$ associates the character $f\mapsto f(x)$ of $\mathscr{C}_0(X)$, and the Gelfand transform of $\mathscr{C}_0(X)$ is then identified with the identity mapping.