Lemma 9

Let A be a commutative star algebra. Every character of A is hermitian and the Gelfand transform is a morphism of star algebras of A into $\mathscr{C}_0(\mathsf{X}(A))$.

It is enough to prove the first assertion (proposition 7 of I, p. 38 and lemma 3 of I, p. 98). Let $\chi$ be a character of A. For every hermitian element $y$ of A, one has $\chi (y) =\mathscr{G}_A(y)(\chi )\in$ Sp$'(y)\subset \mathbf{R}$ (prop. 4 of I, p. 106). Consequently, the character $\chi$ is hermitian (prop. 1 of I, p. 95).