Interactions between von Neumann algebras and quantum algebras

Información del proyecto

INNEQUAL

Identificador del acuerdo de subvención: 702139

DOI

10.3030/702139

Proyecto finalizado el 1 Septiembre 2017

Fecha de la firma de la CE 22 Febrero 2016

Fecha de inicio 1 Septiembre 2016

Fecha de finalización 31 Agosto 2018

Financiado con arreglo a

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Coste total

€ 165 598,80

Aportación de la UE

€ 165 598,80

Coordinado por

UNIVERSITEIT UTRECHT
Netherlands

Periodic Reporting for period 1 - INNEQUAL (Interactions between von Neumann algebras and quantum algebras)

Período documentado: 2016-09-01 hasta 2018-08-31

Resumen del contexto y de los objetivos generales del proyecto

What is the problem/issue being addressed?

The project deals with the structural of q-deformed von Neumann algebras. These are mathematical frameworks to model quantum observables, originally introduced by von Neumann and having several applications in quantum theory and branches of pure mathematics. The project focusses on classification, where in recent years breakthrough results have been obtained for group von Neumann algebras. The proposal brings this classification programme forward by starting the investigation of q-deformed algebras or quantum algebras. Such algebras emerge in quantum gravity and by its formalism generate new classes of von Neumann algebras which in view of this classification programme not much is known.

Why is it important for society?

There is no direct relevance for society, the project deals with pure mathematics.

What are the overall objectives?

1) Obtain classification results for von Neumann algebras
2) Construct new von Neumann algebras of the rare class of `Type III' algebras
3) Applications to geometry and geometric group theory

Trabajo realizado desde el comienzo del proyecto hasta el final del período abarcado por el informe y los principales resultados hasta la fecha

Research output in the period September 2016-August 2017:

2 preprints:
-1 on structural properties of q-deformations (submitted).
-1 on perturbations of commutators (accepted for Annales de l'Institut Fourier).

1 preprint in preparation:
-1 on strong solidity of free quantum groups (submitted in February 2019)

Avances que van más allá del estado de la técnica e impacto potencial esperado (incluida la repercusión socioeconómica y las implicaciones sociales más amplias del proyecto hasta la fecha)

I found new estimates for commutators extending earlier work by myself, coauthors and others. I also found very interesting properties of MASAs in q-Gaussian algebras and obstructions to mutual embeddability for them.

No more results are expected as the project is terminated. I found a permanent (tenure track) job shortly after the project started and therefore had to terminate the grant agreement after 1 year (so when the new job actually started). After continuing research new results have been obtained that address especially the Objectives 1 and 3 mentioned above.

Impact lies in general operator theory and noncommutative analysis. A wider societal impact is not expected.

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