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Groups, Actions and von Neumann algebras

Periodic Reporting for period 4 - GAN (Groups, Actions and von Neumann algebras)

Reporting period: 2018-09-01 to 2020-03-31

"My ERC project ""Groups, Actions and von Neumann algebras"" deals with the structure and the classification of von Neumann algebras arising from free probability theory and group actions on measure spaces. The classification problem for von Neumann algebras is a central question in Operator Algebras and more generally in Functional Analysis."
I developed a deformation/rigidity theory for type III factors analogous to Popa’s deformation/rigidity theory for tracial von Neumann algebras. This aproach has led to the following achievements:
A general unique prime factorization theorem for tensor products of free Araki-Woods factors: Publication 12) ; Conference 8).
A classification theorem for a class of non almost periodic free Araki-Woods factors: Publication 6) ; Conferences 2), 4), 5) .
A strengthened spectral gap criterion for full factors of type III with application to fullness of tensor product factors: Publication 3) ; Conference 1).

A structure theorem for stationary actions of higher rank lattices and higher rank semisimple Lie groups on arbitrary von Neumann algebras (joint work with Rémi Boutonnet). Our results have numerous applications to operator algebras, representation theory, ergodic theory and topological dynamics.
I developed some new tools for the study of type III factors:
A new description of the bicentralizer algebra: Publication 12)
A new and general criterion regarding intertwining subalgebras in arbitrary von Neumann algebras: Publications 5), 12)
A new deformation/rigidity criterion for the unitary conjugacy of faithful normal states on an arbitrary von Neumann algebra: Publication 6).

Our recent joint work with Rémi Boutonnet opens a new chapter in the study of von Neumann algebras associated with (actions of) higher rank lattices. We obtain a non commutative analogue of Nevo-Zimmer's celebrated structure theorem. We plan to continue exploring rigidity aspects of non commutative ergodic theory.