Exercise 3, § 5

Let E be a complex Hilbert space of infinite dimension.

(a) The involutive algebra $\mathscr{C}$alk(E) is a star algebra.

(b) Let $u$ be an element of $\mathscr{L}(E)$ such that $u^*u$ is compact. Now prove that $u$ is compact.

(c) Let G be the group of invertible elements in $\mathscr{C}$alk(E) and let $G_0$ be its neutral component. Let $p: G\rightarrow G/G_0$ be the canonical projection. There exists a unique group isomorphism $\alpha :\mathbf{Z}\rightarrow G/G_0$ such that $\alpha$(ind($u$)) $=p(\pi (u))$ for every Fredholm endomorphism $u$ of E.