Proposition 4

Let A be a star algebra.

a) If A possesses a unit element and if $u$ is a unitary element of A, then Sp($u$)$\subset \mathbf{U}$;

b) If $h$ is a hermitian element of A, then Sp$'(h)\subset \mathbf{R}$.

One may suppose that A is nonzero. Let us prove assertion a). Let $u$ be a unitary element of A. One has $\|u\|=\|u^{-1}\|= 1$ (Remark 1), hence Sp($u$)$\subset \mathbf{U}($I, p. 26, Cor. 3). To prove b), one may suppose that A is unital (Prop. 3). Let $h$ be a hermitian element of A. Then exp($ih$) is unitary (Lemma 5 of I, p. 102). Thus, by Cor. 2 of I, p. 67 and a), one has exp($i$ Sp($h$)) $=$ Sp(exp($ih$))$\subset \mathbf{U}$, which means that Sp($h$)$\subset \mathbf{R}$.