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Suppose $\pi$ is a finite dimensional representation of $C^*$ algebra $A$ on a Hilbert space $H$,then $\pi$ is the direct sum of finite dimensional irreducible subrepresentations.

My quesion is :If $\pi$ is an infinite dimensional representation of $C^*$ algebra $A$ on a Hilbert space $H$ with dim($H$)=$\infty$.Does there exsit a finite dimensional representation of A on $H_0$,where $H_0$ is a subspace of $H$?Can we decompose the infinite dimensional representation?

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Not necessarily. Take $A=B(H)$, and $\pi$ the identity representation.

More generally, take $A$ any C$^*$-algebra with $\pi$ an infinite-dimensional irreducible representation (easy examples of such $A$ are $K(H)$ and UHF$(2^\infty)$). This $\pi$ cannot have summands, since $\pi(A)'=\mathbb C$.

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