# Irreducible representation of product of finite dimensional $C^*$-algebra

Let $(E_{n})_{n\in \Bbb{N}}$ be a sequence of finite-dimensional $C^*$-algebras such that each $E_{n}$ is a direct sum of matrix-blocks with rank at most $k$. Show that every irreducible representation of $\Pi_{n\in\Bbb{N}} E_{n}$ is of dimension$\leq k$.

I know that every irreducible representation of matrix algebra is unitarily equivalent to the identity representation. Moreover, every non-degenerate representation of $A\subseteq K(H)$ is a direct sum of irreducible representations, each of them unitarily equivalent to the identity representation.

Maybe the above observations are not-relevant...any hints or suggestions would be appreciated. Thank you.

Work directly with a sum of blocks. If you think of the restriction of your representation to each of the blocks, it is either zero or $A\longmapsto A\otimes I$. So your representation is (unitarily equivalent to) a direct sum $\bigoplus_n \text{id}\otimes I_{r(n)}$. For the representation to be irreducible, only one summand can appear, and its multiplicity has to be one.
• That $\ker \pi_n$ is zero doesn't tell you what $\pi_n$ is. Only that its image is isomorphic with $E_n$. It may be the case that $\pi(A)=\begin{bmatrix} A&0\\0&A\end{bmatrix}$. Dec 10, 2018 at 23:44
• You have to show that if $B\simeq E_n$, then $B$ is unitarily equivalent to $E_n\otimes I$. It's about multiplicity. Dec 10, 2018 at 23:53