Remark 1

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

Théories spectrales, chapter I, § 6. Algèbres stellaires

Let A be a unital stellar algebra. One has

\|1\|^2=\|1^*1\|=\|1\|

therefore the norm $\|1\|$ is zero or equal to 1. If $A\not=\{0\}$ , one deduces that $\|1\|= 1$ . Consequently, for every unitary element $u$ of A, one has $\|u\|=$ $\|u^*u\|^{1/2}= 1$ .

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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