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Cohomology and $L^2$-Betti numbers for subfactors and quasi-regular inclusions

Presented by: 
Dima Shlyakhtenko University of California, Los Angeles
Wednesday 25th January 2017 - 11:30 to 12:30
INI Seminar Room 1
Co-authors: Sorin Popa (UCLA) and Stefaan Vaes (Leuven)

We introduce L$^2$-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 II$_1$ 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 L$^2$-Betti numbers for the associated equivalence relation. In this common framework, we prove that the L$^2$-Betti numbers vanish for amenable inclusions and we give cohomological characterizations of property (T), the Haagerup property and amenability. We compute the L$^2$-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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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons