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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.

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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.

In other words, an irreducible representation is necessarily of the form "keep one block and erase all the rest".

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  • $\begingroup$ You are welcome :) $\endgroup$ – Martin Argerami May 9 '17 at 23:07
  • $\begingroup$ 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 $\pi(A)=\begin{bmatrix} A&0\\0&A\end{bmatrix}$. $\endgroup$ – Martin Argerami Dec 10 '18 at 23:44
  • $\begingroup$ You have to show that if $B\simeq E_n$, then $B$ is unitarily equivalent to $E_n\otimes I$. It's about multiplicity. $\endgroup$ – Martin Argerami Dec 10 '18 at 23:53

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