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SUMMARY:Cohomology and $L^2$-Betti numbers for subfactors and quasi-regula
 r inclusions - Dima Shlyakhtenko (University of California\, Los Angeles)
DTSTART:20170125T113000Z
DTEND:20170125T123000Z
UID:TALK70271@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:<span>         Co-authors: Sorin Popa (UCLA) and&nbsp\;Stefaan
  Vaes 		(Leuven)<br></span><br>We introduce&nbsp\;L$^2$-Betti numbers\, as
  well as a general homology and cohomology theory for the standard invaria
 nts of subfactors\, through the associated quasi-regular symmetric envelop
 ing inclusion of II$_1$ factors. We actually develop a (co)homology theory
  for arbitrary quasi-regular inclusions of von Neumann algebras. For cross
 ed products by countable groups&nbsp\;&Gamma\;\, we recover the ordinary (
 co)homology of&nbsp\;&Gamma\;. For Cartan subalgebras\, we recover Gaboria
 u&#39\;s&nbsp\;L$^2$-Betti numbers for the associated equivalence relation
 . In this common framework\, we prove that the&nbsp\;L$^2$-Betti numbers v
 anish for amenable inclusions and we give cohomological characterizations 
 of property (T)\, the Haagerup property and amenability. We compute the&nb
 sp\;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.<br>
LOCATION:Seminar Room 1\, Newton Institute
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