Exercise 11, § 1

Let E be a complex Hilbert space. Let $A\subset \mathscr{L}(E)$ be a set of compact endomorphisms such that $u\circ v=v\circ u$ for every pair $(u, v)$ of elements of A and $u^*\in A$ for every $u\in A$.

Now prove that there exists a countable orthonormal family $(e_i)_{i\in I}$ and a family $(\Lambda_i)_{i\in I}$ of mappings $A\rightarrow \mathbf{C}$ such that

for every element $u$ of A and every $x$ in E. If A is a unital subalgebra of $\mathscr{L}$ (E), then $\Lambda_i$ is a character of A for every $i\in I$.