Exercise 11, § 6

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

Spectral Theories, I, p. 181

Let A be an involutive Banach algebra, whose involution satisfies $\\|x^*x\\|=\\|x\\| \\cdot \\|x^*\\|$ for all $x\\in A$ . Then A is a star algebra. (We haveSp $'_A(x^*) =$ Sp $\|y^2\frac{\| = \|}{_{'A}(x)}y\|$ for all $^2$ for $yx$ hermitian, hence $\in A$ , we have therefore $\|y\|=\varrho (y)$ ; observing that

\|x^*\| \cdot \|x\|=\|x^*x\|=\varrho (x^*x)\leqslant \varrho (x^*)\varrho (x) =\varrho (x^2)\leqslant \|x\|^2

whence $\|x^*\|\leqslant \|x\|$ and $\|x^*\|=\|x\|.)$

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