Lemma 3

Tag 02CN, permanent. Cite it as [Bourbaki, Tag 02CN].

Théories spectrales, chapter I, § 6. Algèbres stellaires

Let A be a commutative involutive algebra. The Gelfand transform of A into $\mathscr{C}_0(\mathsf{X}(A))$ is a morphism of involutive algebras if and only if every character of A is hermitian.

In fact, to say that $\mathscr{G}_A$ is a morphism of involutive algebras amounts to saying that, for all $x\in A$ and $\chi \in \mathsf{X}(A)$ , one has

\chi (x^*) =\mathscr{G}_A(x^*)(\chi ) =\mathscr{G}_A(x)(\chi ) =\overline{\chi(x)}

that is to say, that every $\chi$ is hermitian.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Nothing.