Let $A$ be an infinite dimensional $C^*$ algebra.$\psi:A\rightarrow M_{k_1}(\mathbb{C})\oplus M_{k_2}(\mathbb{C})\oplus\ldots\oplus M_{k_r}(\mathbb{C})$

is a nonzero $*$ homomorphism,where $k_1,\ldots k_r$ are positive integers.

I know the fact: every finite dimensional representation is the direct sum of irreducible subrepresentations.

My question is: If $\psi$ is the above representation, does there exist a nonzero irreducible subrepresentaion $\pi:A\rightarrow M_{k_1}(\mathbb{C})$?


No, of course not. Take $A=M_3(\mathbb C)\oplus M_2(\mathbb C)\oplus B (\ell^2 (\mathbb N))$, and $\psi:A\to M_3(\mathbb C)\oplus M_2(\mathbb C)$ given by $$\psi(a\oplus b\oplus T)=(0\oplus b).$$

  • $\begingroup$ Now it is. $\ $ $\endgroup$ – Martin Argerami Sep 14 '18 at 12:13

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