Invariant embeddings and ergodic obstructions

TL;DR

This paper investigates the conditions under which an invariant abelian self-adjoint algebra of operators can be extended to a maximal one that remains invariant under a unitary conjugation, providing positive results in special cases and a surprising counterexample in general.

Contribution

It establishes affirmative results for invariant algebras generated by compact operators and constructs a counterexample showing limitations in the general case.

Findings

Positive extension results for algebras generated by compact operators

Existence of a counterexample in the general case

Insights into invariance properties under unitary conjugation

Abstract

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.