Groups, Actions and von Neumann algebras

Informazioni relative al progetto

GAN

ID dell’accordo di sovvenzione: 637601

DOI

10.3030/637601

Progetto chiuso

Data della firma CE 25 Marzo 2015

Data di avvio 1 Aprile 2015

Data di completamento 31 Marzo 2020

Finanziato da

• EXCELLENT SCIENCE - European Research Council (ERC)

Costo totale

€ 876 750,00

Contributo UE

€ 876 750,00

Coordinato da

UNIVERSITE PARIS-SACLAY
France

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

Periodo di rendicontazione: 2018-09-01 al 2020-03-31

Sintesi del contesto e degli obiettivi generali del progetto

"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."

Lavoro eseguito dall’inizio del progetto fino alla fine del periodo coperto dalla relazione e principali risultati finora ottenuti

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.

Progressi oltre lo stato dell’arte e potenziale impatto previsto (incluso l’impatto socioeconomico e le implicazioni sociali più ampie del progetto fino ad ora)

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.

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