Decomposition of a finite dimensional representation of $C^*$ algebra

Question

Suppose $A$ is a $C^*$ algebra,$\pi$ is a representation of $A$ on finite dimensional Hilbert space $H$,then $\pi=p_1\pi_1\oplus p_2\pi_2\dots\oplus p_n\pi_n$,where $\pi_1,\dots,\pi_n$ are nonequivalent irreducible representations, $p_1,\dots,p_n\in \mathbb{N}$

My quesion is :why the number $p_1$ of equivalent representations of $\pi_1$ is finite?Does there exist infnite many equivalent representations of $\pi_1$?

Answer 1

If you had infinitely many direct sumands, the image would be infinite-dimensional, contradicting your assumption.