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Von Neumann algebras, group actions and discrete quantum groups

Objective

Von Neumann algebras, and more specifically II_1 factors, arise naturally in the study of countable groups and their actions on measure spaces. A central, but extremely hard problem is the classification of these von Neumann algebras in terms of their group/action data. Breakthrough results were recently obtained by Sorin Popa. I presented a combined treatment of these in my Bourbaki lecture notes. In a joint work of Popa and myself, this gave rise to the full classification of certain generalized Bernoulli II_1 factors. In a recent article of mine, it lead for the first time to a family of II_1 factors for which the fusion algebra of finite index bimodules could be entirely computed. Popa's methods open up a wealth of research opportunities. They bring within reach the solution of several long-standing open problems, that constitute the main objectives of the first part of this research proposal: complete descriptions of the finite index subfactor structure of certain II_1 factors, constructions of II_1 factors with a unique group measure space decomposition and computations of orbit equivalence invariants for actions of the free groups. Even approaching these problems would have been completely hopeless just a few years ago. Other constructions of von Neumann algebras arise in the theory of discrete quantum groups. The first rigidity results for quantum group actions on von Neumann algebras constitute the main objective of this second part of the research proposal. Finally, we aim to deal with another connection between quantum groups and operator algebras, through the study of non-commutative random walks and their boundaries. The main originality of this research proposal lies in the interaction between two branches of mathematics: operator algebras and quantum groups. This is clear for the second part of the project and occupies a central place in the first part through subfactor theory.

Call for proposal

ERC-2007-StG
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Host institution

KATHOLIEKE UNIVERSITEIT LEUVEN
Address
Oude Markt 13
3000 Leuven
Belgium
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 500 000
Principal investigator
Stefaan Vaes (Prof.)
Administrative Contact
Stijn Delauré (Dr.)

Beneficiaries (1)

KATHOLIEKE UNIVERSITEIT LEUVEN
Belgium
EU contribution
€ 500 000
Address
Oude Markt 13
3000 Leuven
Activity type
Higher or Secondary Education Establishments
Principal investigator
Stefaan Vaes (Prof.)
Administrative Contact
Stijn Delauré (Dr.)