Cohomology and L2L2-Betti numbers for subfactors and quasi-regular inclusions

58 mins 9 secs, 846.88 MB, MPEG-4 Video 640x360, 29.97 fps, 44100 Hz, 1.94 Mbits/sec

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Description:	Shlyakhtenko, D (University of California, Los Angeles) Wednesday 25th January 2017 - 11:30 to 12:30


Created:	2017-02-08 15:53
Collection:	Operator algebras: subfactors and their applications
Publisher:	Isaac Newton Institute
Copyright:	Shlyakhtenko, D
Language:	eng (English)


Abstract:	Co-authors: Sorin Popa (UCLA) and Stefaan Vaes (Leuven) We introduce L22-Betti numbers, as well as a general homology and cohomology theory for the standard invariants of subfactors, through the associated quasi-regular symmetric enveloping inclusion of II11 factors. We actually develop a (co)homology theory for arbitrary quasi-regular inclusions of von Neumann algebras. For crossed products by countable groups Γ, we recover the ordinary (co)homology of Γ. For Cartan subalgebras, we recover Gaboriau's L22-Betti numbers for the associated equivalence relation. In this common framework, we prove that the L22-Betti numbers vanish for amenable inclusions and we give cohomological characterizations of property (T), the Haagerup property and amenability. We compute the L22-Betti numbers for the standard invariants of the Temperley-Lieb-Jones subfactors and of the Fuss-Catalan subfactors, as well as for free products and tensor products.

Abstract:

Co-authors: Sorin Popa (UCLA) and Stefaan Vaes (Leuven)

We introduce L22-Betti numbers, as well as a general homology and cohomology theory for the standard invariants of subfactors, through the associated quasi-regular symmetric enveloping inclusion of II11 factors. We actually develop a (co)homology theory for arbitrary quasi-regular inclusions of von Neumann algebras. For crossed products by countable groups Γ, we recover the ordinary (co)homology of Γ. For Cartan subalgebras, we recover Gaboriau's L22-Betti numbers for the associated equivalence relation. In this common framework, we prove that the L22-Betti numbers vanish for amenable inclusions and we give cohomological characterizations of property (T), the Haagerup property and amenability. We compute the L22-Betti numbers for the standard invariants of the Temperley-Lieb-Jones subfactors and of the Fuss-Catalan subfactors, as well as for free products and tensor products.

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