Rigidity for von Neumann algebras and their invariants

We give a survey of recent classification results for crossed product von Neumann algebras arising from measure preserving group actions on probability spaces. This includes II_1 factors with uncountable fundamental groups and the construction of W*-superrigid actions where the crossed product entirely remembers the initial group action that it was constructed from.Comment: ICM 2010 Proceedings tex

Type III factors with unique Cartan decomposition

We prove that for any free ergodic nonsingular nonamenable action \Gamma\ \actson (X,\mu) of all \Gamma\ in a large class of groups including all hyperbolic groups, the associated group measure space factor $L^\infty(X) \rtimes \Gamma$ has L^\infty(X) as its unique Cartan subalgebra, up to unitary conjugacy. This generalizes the probability measure preserving case that was established in [PV12]. We also prove primeness and indecomposability results for such crossed products, for the corresponding orbit equivalence relations and for arbitrary amalgamated free products $M_1 *_B M_2$ over a subalgebra B of type I.Comment: v2: we only fixed a LaTeX issu

On Low-Dimensional Locally Compact Quantum Groups

Continuing our research on extensions of locally compact quantum groups, we give a classification of all cocycle matched pairs of Lie algebras in small dimensions and prove that all of them can be exponentiated to cocycle matched pairs of Lie groups. Hence, all of them give rise to locally compact quantum groups by the cocycle bicrossed product construction. We also clarify the notion of an extension of locally compact quantum groups by relating it to the concept of a closed normal quantum subgroup and the quotient construction. Finally, we describe the infinitesimal objects of locally compact quantum quantum groups with 2 and 3 generators - Hopf *-algebras and Lie bialgebras.Comment: 64 pages, LaTeX, needs class-file irmadegm.cls. To appear in Locally Compact Quantum Groups and Groupoids. Proceedings of the Meeting of Theoretical Physicists and Mathematicians, Strasbourg, February 21 - 23, 200

On the optimal paving over MASAs in von Neumann algebras

We prove that if $A$ is a singular MASA in a II$_1$ factor $M$ and $\omega$ is a free ultrafilter, then for any $x\in M\ominus A$, with $\|x\|\leq 1$, and any $n\geq 2$, there exists a partition of $1$ with projections $p_1, p_2, ..., p_n\in A^\omega$ (i.e. a {\it paving}) such that $\|\Sigma_{i=1}^n p_i x p_i\|\leq 2\sqrt{n-1}/n$, and give examples where this is sharp. Some open problems on optimal pavings are discussed.Comment: This paper will appear in a Proceedings Volume for R.V. Kadison's 90th birthda

The boundary of universal discrete quantum groups, exactness and factoriality

We study the C*-algebras and von Neumann algebras associated with the universal discrete quantum groups. They give rise to full prime factors and simple exact C*-algebras. The main tool in our work is the study of an amenable boundary action, yielding the Akemann-Ostrand property. Finally, this boundary can be identified with the Martin or the Poisson boundary of a quantum random walk.Comment: Correction in postal adres