Groups, Actions and von Neumann algebras

Informations projet

GAN

N° de convention de subvention: 637601

DOI

10.3030/637601

Projet clôturé

Date de signature de la CE 25 Mars 2015

Date de début 1 Avril 2015

Date de fin 31 Mars 2020

Financé au titre de

• EXCELLENT SCIENCE - European Research Council (ERC)

Coût total

€ 876 750,00

Contribution de l’UE

€ 876 750,00

Coordonné par

UNIVERSITE PARIS-SACLAY
France

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

Période du rapport: 2018-09-01 au 2020-03-31

Résumé du contexte et des objectifs généraux du projet

"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."

Travail effectué depuis le début du projet jusqu’à la fin de la période considérée dans le rapport et principaux résultats atteints jusqu’à présent

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.

Progrès au-delà de l’état des connaissances et impact potentiel prévu (y compris l’impact socio-économique et les conséquences sociétales plus larges du projet jusqu’à présent)

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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