Bourbaki

Example 2

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

Théories spectrales, chapter I, § 6. Algèbres stellaires

Let X be a locally compact topological space and μ\mu a positive measure on X. The involutive algebra L∞(X,μ)L^{\infty}(X, \mu) (example 2 of I, p. 99) is an involutive Banach algebra, since ∣f∣=∣f‾∣|f|=|\overline{f}| for every element f∈L∞(X,μ)f\in L^{\infty}(X, \mu).

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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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.