Operator Algebras and Group Dynamics – AODynG

We aim to use a dynamical approach. We would like to use certiain dynamical properties to get around algebraic and analytic difficulties. For example we will consider proximaility and its variants, or random walks.

We will also study ubiquitous dynamical systems, such as the action of a group on its space of subgroups, or the action of the unitary group of a von Neumann algebra on the space of its subalgebras. Tools from descriptive set theory may be used.

At this stage we manage to get a result on unitary representations of higher rank lattices by studying some associated operator algebras.

We also proved several dynamical properties of the ubiquitous dynamical systems described above. For example, we classify all the URS of certain families of groups.

A new direction of research, which could be called «non-commutative ergodic theory« seems promising. We hope to be able to further adapt ideas of Margulis in a non-commutative context.

Several paper have been written and submitted for publication.

This project aims to furhter develop the interactions between operator algebras and group geometry and dynamics. This idea of using dynamical tools in the analytical context of operator algebras is rather recent but has already proven efficient, in (at least) two directions, which we suggest to investigate at a deeper level. The first direction is the classification of von Neumann algebras arising from groups and their actions. One of the main questions is to understand the von Neumann algebras arising from higher rank lattices (algebras arising from rank one lattices and their actions being rather well understood by now). The second direction is related to the recent advances on C*-simplicity of groups. We will explore this theme and its connections with topological dynamics, and "stationary" dynamics. The work of Ozawa on solidity gives concrete ways to relate C*-algebras and von Neumann algebras (and topological dynamics). We hope to make further progress in this direction, relying in particular on the recent work of Boutonnet-Ioana-Peterson.
Finally, we will investigate other kinds of questions related to quantum groups or descriptive set theory. These other directions are also based on connections with group dynamics.