Lemma 7

Let A be a stellar algebra.

a) The regular representation $\boldsymbol{\gamma }$ of A (def. 1 of I, p. 16) is isometric, that is to say

$\|x\|=$ sup$_{\|y\|\leqslant 1}\|xy\|$,

for all $x\in A$;

b) For all $x\in A_h$, one has

Let $x\in A$. One has sup$_{\|y\|\leqslant 1}\|xy\|\leqslant \|x\|$. To prove that $\|x\|\leqslant$ sup$_{\|y\|\leqslant 1}\|xy\|$, one may suppose that $\|x\|= 1$. Then the element $y$ = $x^*$ satisfies $\|y\|=\|x^*\|= 1$, and $\|xy\|=\|x\|^2= 1$, whence assertion a).

Suppose that $x$ is hermitian. Then $\|x^2\|=\|x^*x\|=\|x\|^2$, whence by induction $\|x^{2^n}\|^{2^{-n}}=\|x\|$ for every integer $n\geqslant 1$, whence assertion b) by prop. 1 of I, p. 20.