Bourbaki

Definition 5

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

Théories spectrales, chapter I, § 6. Algèbres stellaires

The involutive morphism f↦(GB−1∘φx)(f)f\mapsto (\mathscr{G}_B^{-1}\circ \varphi_x)(f) of C\mathscr{C}(SpA(x)_A(x)) into A is called the mapping of continuous functional calculus of xx in A. It is denoted by f↦f(x)f\mapsto f(x).

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.

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Cites

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Machine translation of the English, by gpt-5-6-mini, gpt-5.4, against glossary version 34. Not checked by a person. That model is a cut down version of the one the rest of this language was written by.