Let $H$ denote a Hilbert space and let $\cal A$ be a subalgebra of the algebra ${\cal B}(H)$ of all bounded operators on $H$ such that $\cal A$ consists of compact operators only and such that each vector $v\in H$ lies in the closure of ${\cal A}v$.

Is it true that there must exist an irreducible subspace for $\cal A$?

properirreducible subspaces, but that was my mistaken reading. Thanks for clarification – Yemon Choi Oct 24 '12 at 11:55