Exercise 10, § 1

Let $E$ be a complex Hilbert space. Let $A$ be a star-algebra of $\mathscr{L}(E)$. Suppose that the representation of A in E is irreducible and that A contains a nonzero compact endomorphism.

(a) The star-algebra A contains a nonzero projector of finite rank. (Now prove that A contains a nonzero compact hermitian endomorphism and apply the functional calculus.)

(b) The star-algebra A contains a projector of rank 1. (If $p\in A$ is a nonzero projector of minimal rank, prove that the algebra $pAp$ is contained in $\mathbf{C}p.$)

(c) Deduce that A contains $\mathscr{L}^c(E)$.

(d) Conversely, every star-algebra of $\mathscr{L}(E)$ containing $\mathscr{L}^c(E)$ is irreducible.