Bourbaki

Lemma 1

Tag 024R, permanent. Cite it as [Bourbaki, Tag 024R].

Théories spectrales, chapter I, § 1. Spectres et caractères

Let A be an algebra. For every idempotent jj of A, the subspace jAjjAj of A is the set of x∈Ax\in A such that xj=jx=xxj=jx=x. It is a subalgebra of A having jj as unit element.

The proof is elementary.

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

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Cites

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Machine translation of the English, by gpt-5.4, gpt-5-6-mini, 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.