Classification of group von Neumann algebras
January 29 to February 2, 2018
American Institute of Mathematics,
San Jose, California
and Stefaan Vaes
This workshop 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
- 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?
Material from the workshop
A list of participants.
The workshop schedule.
A report on the workshop activities.
A list of open problems.