Rigidity and classification of von Neumann algebra.. (RIGIDITY)
Rigidity and classification of von Neumann algebras
Start date: Jul 1, 2014,
End date: Jun 30, 2019
"Sorin Popa's deformation/rigidity theory has lead to an enormous progress in our understanding of von Neumann algebras coming from discrete groups and their actions on probability spaces. In a five year long collaboration with Sorin Popa, we solved many long-standing open problems in this area, including superrigidity theorems for group measure space II_1 factors, results on the possible fundamental groups of II_1 factors, and uniqueness theorems for Cartan subalgebras.In the first part of the project, we want to establish new unique Cartan decomposition theorems for II_1 factors coming from hitherto intractable groups. Using methods coming from Lie groups, ergodic theory and geometric group theory, we want to reach such results for lattices in higher rank simple Lie groups, and for countable groups with nonvanishing L^2-Betti numbers. An important intermediate step will be the unique Cartan decomposition of Bernoulli crossed products.Secondly we want to prove classification theorems for type III factors that are equally strong as the existing results for the type II_1 case. This includes a complete classification of the noncommutative Bernoulli shifts of the free groups and will require an intricate combination of Tomita/Takesaki and deformation/rigidity theory.The methods developed so far bring within reach an attack on two of the most important open problems in operator algebras and functional analysis: the free group factor problem and Connes's rigidity conjecture. The exact progress on these problems is of course unforeseeable, but it is sure that the research on these problems will lead to an even deeper interaction between diverse areas of mathematics as operator algebras, group theory, functional analysis, ergodic theory, and descriptive set theory. Intermediate goals are the classification of natural classes of group von Neumann algebras, including those coming from Baumslag-Solitar groups, wreath product groups, and other families of discrete groups."
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