Ergodic Theory and Maximal Abelian Subalgebras of the Hyperfinite Factor
✍ Scribed by Sergey Neshveyev; Erling Størmer
- Publisher
- Elsevier Science
- Year
- 2002
- Tongue
- English
- Weight
- 245 KB
- Volume
- 195
- Category
- Article
- ISSN
- 0022-1236
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✦ Synopsis
Let T be a free ergodic measure-preserving action of an abelian group G on ðX ; mÞ: The crossed product algebra R T ¼ L 1 ðX ; mÞs G has two distinguished masas, the image C T of L 1 ðX ; mÞ and the algebra S T generated by the image of G: We conjecture that conjugacy of the singular masas S T ð1Þ and S T ð2Þ for weakly mixing actions T ð1Þ and T ð2Þ 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 g : R T ð1Þ ! R T ð2Þ such that gðS T ð1Þ Þ ¼ S T ð2Þ has the property that the Cartan subalgebras gðC T ð1Þ Þ and C T ð2Þ of R T ð2Þ are inner conjugate. We discuss a stronger conjecture about the structure of the automorphism group AutðR T ; S T Þ; 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. # 2002 Elsevier Science (USA)