Suppose $A$ is a non-unital $C^*$ algebra,can we conclude that there must exist a nonzero finite irreducible representation of $A$.
2026-03-25 23:42:31.1774482151
existence of finite irreducible reprentation of a nonunital $C^*$ algebra
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We cannot guarantee the existence of such a representation. Consider $K(H)$, for $H$ a separable infinite-dimensional Hilbert space. This is non-unital and simple, so any representation of this has to be infinite-dimensional.