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Post-edited  Cohomology and $L^2$-Betti numbers for subfactors and quasi-regular inclusions Vaes, Stefaan (Auteur de la Conférence) | CIRM (Editeur )

I present a joint work with S. Popa and D. Shlyakhtenko introducing a cohomology theory for quasi-regular inclusions of von Neumann algebras. In particular, we define $L^2$-cohomology and $L^2$-Betti numbers for such inclusions. Applying this to the symmetric enveloping inclusion of a finite index subfactor, we get a cohomology theory and a definition of $L^2$-Betti numbers for finite index subfactors, as well as for arbitrary rigid $C^*$-tensor categories. For the inclusion of a Cartan subalgebra in a $II_1$ factor, we recover Gaboriau's $L^2$-Betti numbers for equivalence relations. I present a joint work with S. Popa and D. Shlyakhtenko introducing a cohomology theory for quasi-regular inclusions of von Neumann algebras. In particular, we define $L^2$-cohomology and $L^2$-Betti numbers for such inclusions. Applying this to the symmetric enveloping inclusion of a finite index subfactor, we get a cohomology theory and a definition of $L^2$-Betti numbers for finite index subfactors, as well as for arbitrary rigid $C^*$-tensor ...

Multi angle  Approximating freeness under constraints, with applications Popa, Sorin (Auteur de la Conférence) | CIRM (Editeur )

I will discuss a method for constructing a Haar unitary $u$ in a subalgebra $B$ of a $II_1$ factor $M$ that's "as independent as possible" (approximately) with respect to a given finite set of elements in $M$. The technique consists of "patching up infinitesimal pieces" of $u$. This method had some striking applications over the years:
1. vanishing of the 1-cohomology for $M$ with values into the compact operators (1985);
2. reconstruction of subfactors through amalgamated free products and axiomatisation of standard invariants (1990-1994).
3. first positive results on Kadison-Singer type paving (2013);
4. vanishing of the continuous version of Connes-Shlyakhtenko 1-cohomology (with Vaes in Jan. 2014) and of smooth 1-cohomology (with Galatan in June 2014).
I will discuss a method for constructing a Haar unitary $u$ in a subalgebra $B$ of a $II_1$ factor $M$ that's "as independent as possible" (approximately) with respect to a given finite set of elements in $M$. The technique consists of "patching up infinitesimal pieces" of $u$. This method had some striking applications over the years:
1. vanishing of the 1-cohomology for $M$ with values into the compact operators (1985);
2. reconstruction of ...

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