We consider the following question: Let $\mathcal{ A}$ be an abelian self-adjoint algebra of bounded operators on a Hilbert space $\mathcal{ H}$. Assume that $\mathcal{ A}$ is invariant under conjugation by a unitary operator $U$, i.e., $U^* AU$ is in $\mathcal{ A}$ for every member $A$ of $\mathcal{ A}$. Is there a maximal abelian self-adjoint algebra containing $\mathcal{ A}$, 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 $\mathcal{ A}$ is generated by a compact operator. We also construct a counterexample in the general case, whose existence is perhaps surprising.
Categories: math.FA