Ergodic Theory and Maximal Abelian Subalgebras of the Hyperfinite Factor

Abstract

Let T be a free ergodic measure-preserving action of an abelian group G on ( X,μ). The crossed product algebra R T = L ∞( X,μ)⋊ G has two distinguished masas, the image C T of L ∞( X,μ) and the algebra S T generated by the image of G. We conjecture that conjugacy of the singular masas S T (1) and S T (2) for weakly mixing actions T (1) and T (2) of different groups implies that the groups are isomorphic and the actions are conjugate with respect to this isomorphism. Our main result supporting this conjecture is that the conclusion is true under the additional assumption that the isomorphism γ : R T (1) → R T (2) such that γ( S T (1) )= S T (2) has the property that the Cartan subalgebras γ( C T (1) ) and C T (2) of R T (2) are inner conjugate. We discuss a stronger conjecture about the structure of the automorphism group Aut( R T,S T ), and a weaker one about entropy as a conjugacy invariant. We study also the Pukanszky and some related invariants of S T , and show that they have a simple interpretation in terms of the spectral theory of the action T. It follows that essentially all values of the Pukanszky invariant are realized by the masas S T , and there exist non-conjugate singular masas with the same Pukanszky invariant.