Remark 2

Tag 0255, permanent. Cite it as [Bourbaki, Tag 0255].

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

It follows from Remark 1 that, if B is an algebra over K and if $x\in B$ , one has:

Sp $'_B(x) =$ Sp $_{\widetilde{B}}(x) =$ Sp $_{\widetilde{B}}(x)\cup \{0\}=$ Sp $'_{\widetilde{B}}(x)$ .

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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