Proposition 7

Let A be a commutative Banach algebra.

a) The Gelfand transform $\mathscr{G}$ defines a morphism of A into $\mathscr{C}_0(\mathsf{X}(A))$ such that $\|\mathscr{G}(x)\|=\varrho (x)\leqslant \|x\|$ for every $x\in A$;

b) In order that the Gelfand transform $\mathscr{G}$ be isometric, it is necessary and sufficient that $\|x^2\|=\|x\|^2$ for every $x\in A$.

The mapping $\mathscr{G}$ is a morphism of A into $\mathscr{C}_0(\mathsf{X}(A))$ by No. $^o7$ of I, p. 9 and Prop. 5 of I, p. 37, and satisfies $\|\mathscr{G}(x)\|=\varrho (x)$ by Prop. 6 and Cor. 5 of I, p. 28. Assertion b) follows from a) and Remark 1 of I, p. 21.