Invariant embeddings and ergodic obstructions

We consider the following question: Let A be an abelian self-adjoint algebra of bounded operators on a Hilbert space H. Assume that A is invariant under conjugation by a unitary operator U, i. e. , U^* AU is in A for every member A of A. Is there a maximal abelian self-adjoint algebra containing A, which is still invariant under conjugation by U? The answer, which is easily seen to be yes in finite dimensions, is not trivial in general. We prove affirmative answers in special cases including the one where A is generated by a compact operator. We also construct a counterexample in the general case, whose existence is perhaps surprising.