TAILIEUCHUNG - Đề tài " On a class of type II1 factors with Betti numbers invariants "

We prove that a type II1 factor M can have at most one Cartan subalgebra A satisfying a combination of rigidity and compact approximation properties. We use this result to show that within the class HT of factors M having such Cartan subalgebras A ⊂ M , the Betti numbers of the standard equivalence relation associated with A ⊂ M ([G2]), are in fact isomorphism invariants for HT the factors M , βn (M ), n ≥ 0. The class HT is closed under amplifications HT HT and tensor products, | Annals of Mathematics On a class of type II1 factors with Betti numbers invariants By Sorin Popa Annals of Mathematics 163 2006 809 899 On a class of type III factors with Betti numbers invariants By Sorin Popa Abstract We prove that a type III factor M can have at most one Cartan subalgebra A satisfying a combination of rigidity and compact approximation properties. We use this result to show that within the class HT of factors M having such Cartan subalgebras A c M the Betti numbers of the standard equivalence relation associated with A c M G2 are in fact isomorphism invariants for the factors M 3n M n 0. The class HT is closed under amplifications and tensor products with the Betti numbers satisfying 3n M 3n M t Yt 0 and a Kunneth type formula. An example of a factor in the class HT is given by the group von Neumann factor M L Z2 X SL 2 Z for which 3ht M 31 SL 2 Z 1 12. Thus M M Yt 1 showing that the fundamental group of M is trivial. This solves a long standing problem of R. V. Kadison. Also our results bring some insight into a recent problem of A. Connes and answer a number of open questions on von Neumann algebras. Contents 0. Introduction 1. Preliminaries . Pointed correspondences . Completely positive maps as Hilbert space operators . The basic construction and its compact ideal space . Discrete embeddings and bimodule decomposition 2. Relative Property H Definition and examples 3. More on property H 4. Rigid embeddings Definitions and properties 5. More on rigid embeddings 6. HT subalgebras and the class HT 7. Subfactors of an HT factor 8. Betti numbers for HT factors Appendix Some conjugacy results Supported in part by a NSF Grant 0100883. 810 SORIN POPA 0. Introduction We consider in this paper the class of type III factors with maximal abelian -subalgebras satisfying both a weak rigidity property in the spirit of Kazhdan Margulis Ka Ma and Connes-Jones CJ and a weak amenability property in the spirit of Haagerup s compact approximation .

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