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ERC

GAN Report Summary

Project ID: 637601
Funded under: H2020-EU.1.1.

Periodic Reporting for period 3 - GAN (Groups, Actions and von Neumann algebras)

Reporting period: 2017-03-01 to 2018-08-31

Summary of the context and overall objectives of the project

"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."

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

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).

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

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).
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