Joint Mathematics Colloquium (ICMAT-UAM-UC3M-UCM)
Rigidity theory for von Neumann algebras
Speaker: Stefaan Vaes (KU-Leuven, Belgium)Date: Thursday, 22 September 2022 - 11:30Place: Aula Naranja, ICMAT
Online: https://youtu.be/o2P0MB98ims
Abstract:
Discrete groups and their actions on probability spaces give rise to algebras \(M\) of Hilbert space operators. The concept of amenability for groups, discovered by von Neumann in his explanation of the Banach-Tarski paradox, leads to a sharp dichotomy for these II\(_1\) factors \(M\). If the group is amenable, \(M\) is always isomorphic to the unique hyperfinite II\(_1\) factor, by Connes' theorem. When the group is nonamenable, Popa's deformation/rigidity theory leads to striking rigidity results. This includes W\(^*\)-superrigidity theorems where the group and its action can be entirely recovered from the ambient II\(_1\) factor. I will give a survey of some of these results, including the challenging problem to decide when a given II\(_1\) factor can be embedded into another one.
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