Proposition 5

Let A be a commutative Banach algebra. For every $x\in A$, the function $\mathscr{G}(x)$ belongs to the commutative Banach algebra $\mathscr{C}_0(\mathsf{X}(A))$ of continuous functions on $\mathsf{X}(A)$ tending to 0 at infinity.

By definition (cf. n$^o7$ of I, p. 9), the function $\mathscr{G}'_A(x) :\chi \mapsto \chi (x)$ is continuous on $\mathsf{X}'(A)$ and zero at 0. Since $\mathsf{X}'(A)$ is identified with the Alexandroff compactification of $\mathsf{X}(A)$ by cor. 1 of I, p. 29, the proposition follows.