JCJC SIMI 1 - JCJC - SIMI 1 - Mathématiques et interactions

Von Neumann algebras: structure, classification, rigidity and applications – NEUMANN

NEUMANN

Von Neumann algebras: structure, classification, rigidity and applications

Context, position and objectives of the proposal

We investigate von Neumann algebras arising from group theory, ergodic theory of group actions and free probability theory. To any probability measure-preserving action of a countable group on the standard probability space, one can associate the so-called group measure space construction of Murray and von Neumann. One of the most important questions regarding this construction is the following: how much information of the group/action data does the von Neumann algebra remember?<br /><br />When the group is amenable, Connes' celebrated result shows that the von Neumann algebra only remembers the amenability, and forgets all the information about the group and the action. In the nonamenable case, the situation is far more complex and rich. The introduction by S. Popa of his Deformation/Rigidity Theory ten years ago allowed to prove striking rigidity/classification results regarding the group measure-space construction. This is a very active field of research nowadays in operator algebras and functional analysis.<br /><br />Many important problems addressed by S. Popa have not yet been solved. We plan to work on these questions and other open problems relating von Neumann algebras to ergodic theory, lattices in semisimple Lie groups, percolation on nonamenable infinite graphs, noncommutative probability theory, to name a few.

Each calendar year, a two-day meeting will be organized. This meeting will bring together all the participants to this project to discuss their current ongoing research, the problems they are thinking on (especially those that they are stuck on) in order to foster co-operation and synergy. These meetings are meant to be informal and all participants will be able to present their research. They will take place alternatively in Caen, Paris and Lyon.

We plan to organize an international four-day workshop to be held at UMPA, ENS Lyon, during the third year of this project. We will invite three plenary lecturers from abroad to give a mini-course (three hours each) on different topics involving von Neumann algebras, representation group theory or free probability theory. We will invite as well a few speakers to deliver a 50 min-talk on their most recent ongoing research. Young researchers (PhD students and post-doctoral fellows) working in the field will be strongly encouraged to participate to the workshop.

We already obtained several new and significant results.

We aim at applying von Neumann algebras theory to open questions from other fields.

We investigate von Neumann algebras arising from group theory, ergodic theory of group actions and free probability theory. To any probability measure-preserving action of a countable group on the standard probability space, one can associate the so-called group measure space construction of Murray and von Neumann. One of the most important questions regarding this construction is the following: how much information of the group/action data does the von Neumann algebra remember?
When the group is amenable, Connes' celebrated result shows that the von Neumann algebra only remembers the amenability, and forgets all the information about the group and the action. In the nonamenable case, the situation is far more complex and rich. The introduction by S. Popa of his Deformation/Rigidity Theory ten years ago allowed to prove striking rigidity/classification results regarding the group measure-space construction. This is a very active field of research nowdays in operator algebras and functional analysis.
Many important problems addressed by S. Popa have not yet been solved. We plan to work on these questions and other open problems relating von Neumann algebras to ergodic theory, lattices in semisimple Lie groups, percolation on nonamenable infinite graphs, noncommutative probability theory, to name a few.
To achieve these advances in the structure, classification and rigidity of von Neumann algebras, we will combine complementary skills and techniques from three mathematicians specialized and well trained in deformation/rigidity techniques, one mathematician specialized in free probability theory and one in operator space theory.

Project coordinator

Monsieur Cyril HOUDAYER (Unité de Mathématiques Pures et Appliquées, Ecole Normale Supérieure de Lyon) – cyril.houdayer@math.u-psud.fr

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

Partner

UMPA, ENS Lyon Unité de Mathématiques Pures et Appliquées, Ecole Normale Supérieure de Lyon

Help of the ANR 89,995 euros
Beginning and duration of the scientific project: August 2012 - 36 Months

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