I am having trouble with the following proof:

Prove: Every non-degenerate representation $\{ \pi , H \} $ of an involutive Banach algebra is a direct sum of cyclic representations.



a representation $\{ \pi , H\}$ of an involutive Banach algebra $A$ is an involutive algebra homomrphism $\pi : A\rightarrow \mathcal{B}(H)$ for some Hilbert space $H$

$\{ \pi , H \}$ is called non-degenerate if $\overline {span}\{ \pi (x) \xi : x\in A , \xi\in H \} = H$ (or equivalently, if for every $\xi \in H$, there is an $x \in A$ such that $\pi (x) \xi \not = 0$).

$\{ \pi , H \} $ is called cyclic if there exists $\xi \in H$ such that $\overline {\pi (A) \xi } = H$.


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