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In the world of von Neumann algebras, factors can be classified into three types. The type III factors are those that do not have a trace. They are related to nonsingular ergodic actions, regular representations of non-unimodular groups and quantum field theory. Some of the key structural properties of this class of factors are still not well understood. In this mini-course, I will give a gentle introduction to the theory of type III factors and to the deepest open problem in the theory : Connes's Bicentralizer Problem (1979).[-]
In the world of von Neumann algebras, factors can be classified into three types. The type III factors are those that do not have a trace. They are related to nonsingular ergodic actions, regular representations of non-unimodular groups and quantum field theory. Some of the key structural properties of this class of factors are still not well understood. In this mini-course, I will give a gentle introduction to the theory of type III factors and ...[+]

46L10 ; 46L30 ; 46L36 ; 46L37 ; 46L55

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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 ...[+]

46L37 ; 46L10

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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 ...[+]

46L10 ; 46L37

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