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,mu). The crossed product algebra R_T has two distinguished masas, the image C_T of L^infty(X,mu) 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 gamma of R_{T^(1)} onto R_{T^(2)} such that gamma(S_{T^(1)})=S_{T^(2)} has the property that the Cartan subalgebras gamma(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.