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Interactions between von Neumann algebras and quantum algebras

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

Reporting period: 2016-09-01 to 2018-08-31

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
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)
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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