Classification of group von Neumann algebras

Classification of group von Neumann algebras

This workshop, sponsored by AIM and the NSF, will be devoted to developing new ideas and approaches to the classification of group and group-action von Neumann algebras. The workshop will bring together researchers with deep expertise in von Neumann algebras, geometric group theory, and Lie groups, as well as young researchers, to cooperate intensively on some of the outstanding open problems in the field, such as Connes' rigidity conjecture, while also exploring new avenues of research and interactions among these disciplines.

At a March 2009 workshop at UCLA, Sorin Popa proposed an ambitious list of open problems in group and group-action von Neumann algebras which has since steered the direction of the field. This initial list of problems was attacked with a flurry of definitive results, leading Popa to revisit the list in 2013. The workshop will allow an opportunity to reassess the state of the field in the wake of a string of even more recent successes, to catalyze new directions in research and to provide an updated vision for the field.

The topics of the workshop are the following three outstanding problems appearing on Popa's lists:

• If two infinite conjugacy class countable discrete property (T) groups have (stably) isomorphic group von Neumann algebras does this imply that the groups are isomorphic? This was originally conjectured to hold by Connes in the late 1970s.
• If two diffuse, amenable subalgebras of a free group factor have diffuse intersection, is the algebra generated by both subalgebras still amenable?
• For any nonamenable group does the group-measure space von Neumann algebra associated to a Bernoulli action have unique Cartan subalgebra? If the group has nonzero $\ell^2$-quasicohomology, does every associated group-measure space von Neumann algebra have unique Cartan subalgebra?

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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