Exercise 3, § 2

Let $S\subset \mathbf{C}$ be a compact subset of $\mathbf{C}$. Let $\varphi :\mathscr{C}(S)\rightarrow \mathscr{L}(E)$ be a continuous unital morphism of involutive algebras. There exists a unique normal endomorphism $u$ of E whose spectrum is S and such that $\varphi (f) =f(u)$ for every $f\in \mathscr{L}^{\infty}(S)$.