Irreducible algebras of operators which contain a minimal idempotent.

Abstract:

We prove that when A is a closed subalgebra of the bounded operators on a reflexive Banach space X, which acts irreducibly on X and contains a minimal idempotent, then every bounded operator with finite dimensional range on X is in A. We use this result to prove that every continuous irreducible representation of a GCR-algebra on a Hilbert space $\mathcal {H}$ is similar to a $^ \ast$-representation on $\mathcal {H}$.