Project description
The Baum–Connes conjecture for discrete quantum groups exhibiting torsion phenomena
The Baum–Connes conjecture suggests that the K-theory of the reduced C-star algebra of a group is identical with the equivariant K-homology of a certain sort of classifying space for the group. Meyer and Nest have reformulated the conjecture using the language of triangulated categories. This reformulation works for both classical locally compact groups and torsion-free discrete quantum groups. The EU-funded CONCOQUANT project aims to address key questions regarding torsion phenomena in discrete quantum groups. The project will also study how to obtain stability properties of Baum–Connes for relevant constructions of quantum groups – quantum semi-direct products and free wreath products.
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.
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Funding Scheme
MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Coordinator
1165 Kobenhavn
Denmark