When $H$ is separable infinite dimensional,$K(H)$ has no finite representation.Do there exist other non-unital $C^*$ algebras such that their representations cannot be finite dimensional?
2026-03-26 17:12:04.1774545124
infinite dimensional representation of a $C^*$ algebra
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Yes. Any simple infinite-dimensional C$^*$-algebra (that is, "most" of them in the sense that these are precisely the ones people study a lot) does not admit a finite-dimensional representation.