Lemma 6

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

Théories spectrales, chapter I, § 6. Algèbres stellaires

Let A be a Banach algebra endowed with an involution satisfying

\|x\|^2\leqslant \|x^*x\| \tag{1}

for every $x\in A$ . Then A is a stellar algebra.

Let $x\in A$ . Then $\|x\|^2\leqslant \|x^*\| \cdot \|x\|$ , whence $\|x\|\leqslant \|x^*\|$ . By interchanging the roles of $x$ and $x^*$ , one sees that $\|x\|=\|x^*\|$ . Thus $\|x^*x\|\leqslant$ $\|x^*\|\|x\|\leqslant \|x\|^2$ and the hypothesis implies the equality $\|x\|^2=\|x^*x\|$ .

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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