Theorem 1

Let A be a commutative star algebra. The Gelfand transform is an isometric isomorphism of the star algebra A onto the star algebra $\mathscr{C}_0(\mathsf{X}(A))$ of continuous functions on $\mathsf{X}(A)$ tending to 0 at infinity.

The Gelfand transform is a morphism of involutive algebras of A into $\mathscr{C}_0(\mathsf{X}(A))$ (lemma 9). Let B be its image. It is an involutive subalgebra of $\mathscr{C}_0(\mathsf{X}(A))$. The elements of B separate the points of $\mathsf{X}(A)$ by definition. Let $\chi \in \mathsf{X}(A)$. There exists $x\in A$ such that $\chi (x)\not= 0$, hence $f=\mathscr{G}(x)$ is an element of B such that $f(\chi )\not= 0$. By TG, X, p. 40, cor. 2, the subalgebra B is therefore dense in $\mathscr{C}_0(\mathsf{X}(A))$.

For every hermitian element $y$ of A, one has $\|\mathscr{G}(y)\|=\varrho (y) =\|y\|$ (prop. 7 of I, p. 38 and formula (2) of I, p. 104), whence, for every $x\in A$, the equalities

Thus $\mathscr{G}$ is isometric, and its image B is therefore closed (lemma 8). One concludes that $B =\mathscr{C}_0(\mathsf{X}(A))$.