CONCOQUANT · Connes' Conjectures with Quantum Groups
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2020-10-01 → 2022-09-30
- EU contribution
- €219,312
- Participants
- 1
- Scheme
- MSCA-IF
Lines connect the coordinator with its partners.
Results in brief
Connes' Conjectures with Quantum Groups
For the past few decades, the Baum-Connes conjecture has been a driving force in understanding the structure of group C*-algebras with diverse techniques coming from geometry, analysis, representation theory and, more recently, category theory. There is no counter example for its original formulation, but the one with coefficients has been revealed to be false thanks to the work of N. Higson, V. Lafforgue and G. Skandalis. For this reason we refer to it as the Baum-Connes property (BC property for short). R. Meyer and R. Nest have developed a categorical framework for the BC property avoiding any explicit geometrical construction, which allows to obtain a formulation for torsion-free discrete quantum groups. My project has been focused on the BC property formulation for discrete quantum groups (not necessarily torsion-free), which has been an open problem since the beginning of the 2000's.One of the main questions that the project has aimed to understand is the torsion phenomenon for discrete quantum groups in relation with the categorical framework of Meyer-Nest. The project has also aimed to carry out further developments in the quantum setting. On the one hand, I have studied the notion of Künneth formula in the framework of quantum groups. In algebraic topology, a Künneth formula allows to relate the homology of two objects to the homology of their product. In the framework of operator algebras, we are interested in determing the K-theory of a tensor product of C*-algebras in terms of the K-theory of each of the C*-algebras involved. In this respect, a landmark work was done by J. Rosenberg and C. Schochet in the end of the 80's. Moreover, Künneth formulas turn out to be key tools for understanding different aspects of C*-algebras such as classification problems. On the other hand, I have addressed relevant open questions concerning the Connes' Embedding property (CEP for short). Generally speaking, CEP asks whether it is possible to approximate some kind of von Neumann algebras using finite-dimensional data. It is an outstanding problem in operator algebras with connections with multiple branches of mathematics such as functional analysis or group theory.
Data: CORDIS, © European Union
Project objective
This project focuses on the Baum-Connes conjecture formulation for discrete quantum groups. The work of R. Meyer and R. Nest in the second half of 2000's has lead to a categorial formulation of the Baum-Connes conjecture in the context of triangulated categories. This reformulation works for both classical locally compact groups and torsion-free discrete quantum groups. Thus one of the main questions that the project aims to understand is the torsion phenomena for discrete quantum groups in relation with the categorical framework of Meyer-Nest. This will allow to manipulate conveniently the corresponding homological algebra for two main purposes. First, introducing a new insight for a proper formulation of the Baum-Connes conjecture for arbitrary discrete quantum groups. Second, carrying out explicit K-theory computations of C*-algebras defining relevant examples of quantum semi-direct products and free wreath products. The compact bicrossed product construction will be studied in detail in this framework in order to classify its torsion actions and to obtain the corresponding stability result of BC. Moreover, this construction will provide a vast class of new examples satisfying the quantum BC conjecture coming from recent constructions by several authors involving approximation properties such as property (T) or Haagerup property. The project aims also to carry out further developments in the quantum setting. One the one hand, defining and developping a quantum equivariant Künneth formula theory using the notion of Künneth functor. On the other hand, studying the recently discovered connections between compact quantum groups and non-local games, in the framework of quantum information theory, in order to address relevant open questions concerning the Connes' embedding conjecture with potential applications and consequences within the area of algorithm theory in computer science.
Original text from CORDIS.
Participants
- KOBENHAVNS UNIVERSITET · KOBENHAVNCoordinatorDenmark
Links
Data: CORDIS, © European Union