Lemma 5

Let A be an involutive Banach algebra.

a) For every $x\in A$, one has exp($x$)$^*=$ exp($x^*$) ;

b) Let $x\in A_h$ be a hermitian element. Then exp($ix$) is unitary.

In fact, since the involution on A is continuous, one has

exp($x$)$^*=(\sum_{n=0}^{\infty}xn^n$! $)^*=\sum_{n=0}^{\infty}(x\overset{*}{n}$!$)^n=$ exp($x^*$)

for every $x\in A$ (formula (9) of I, p. 78). If $x\in A_h$, there follows

exp($ix$)$^*=$ exp(($ix$)$^*$) $=$ exp($-ix$) $=$ exp($ix$)$^{-1}$

(formula (11) of I, p. 78).