Periodic Reporting for period 1 - CONCOQUANT (Connes' Conjectures with Quantum Groups)
Periodo di rendicontazione: 2020-10-01 al 2022-09-30
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.
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.
The study of the torsion phenomenon for discrete quantum groups has allowed me to manipulate conveniently the corresponding homological algebra for two main purposes.
On the one hand, I have introduced a new insight for a proper formulation of the BC property for arbitrary discrete quantum groups. Moreover, this research has yield two supplementary results not contemplated in the original project. First, I have solved the "cleftness" problem for compact quantum groups, which was an open problem in the subject. Generally speaking, it states that a twisted irreducible finite-dimensional representation of a compact quantum group is equivalent to a torsion action of the quantum group. This result allows thus to study certain torsion phenomenon for compact quantum groups from a representation theoretic point of view. Second, I have realised a connection between the quantum BC property and the theory of quantum groupoids, which is innovative in the subject.
On the other hand, I have started explicit K-theory computations of C*-algebras defining relevant examples of quantum groups; namely, "quantum semi-direct products". In particular, I have given a classification of its torsion phenomenon. I have also studied the case of profinite (quantum) groups in relation with the BC property. Namely, I have showed that duals of compact groups (not necessarily connected) satisfy the BC property. This complements a previous result by R. Meyer and R. Nest. For this I have developed a reformulation of the equivariant Kasparov in terms of explicit formulae using the representation theory of a given compact quantum group. This reformulation turns out to be a key technical tool to address the general construction of compact bicrossed products in relation with approximation properties, which is now part of my future research project.
On a different note, I have defined a quantum equivariant Künneth formula using the notion of Künneth functor in order to relate the quantum BC property to the Künneth formula.
Due to their complexity, some problems have not been solved as expected. For instance, this is is the case for the problems addressed around CEP. More time and further investigation is needed. I have studied the recently discovered connections between compact quantum groups and non-local games in the framework of quantum information theory in order to address relevant questions around CEP and quantum strategies for the graph isomorphism game. In the course of the action, however, I have acquired new directions to address these problems by discovering supplementary open questions related to my original objective.
On the one hand, I have introduced a new insight for a proper formulation of the BC property for arbitrary discrete quantum groups. Moreover, this research has yield two supplementary results not contemplated in the original project. First, I have solved the "cleftness" problem for compact quantum groups, which was an open problem in the subject. Generally speaking, it states that a twisted irreducible finite-dimensional representation of a compact quantum group is equivalent to a torsion action of the quantum group. This result allows thus to study certain torsion phenomenon for compact quantum groups from a representation theoretic point of view. Second, I have realised a connection between the quantum BC property and the theory of quantum groupoids, which is innovative in the subject.
On the other hand, I have started explicit K-theory computations of C*-algebras defining relevant examples of quantum groups; namely, "quantum semi-direct products". In particular, I have given a classification of its torsion phenomenon. I have also studied the case of profinite (quantum) groups in relation with the BC property. Namely, I have showed that duals of compact groups (not necessarily connected) satisfy the BC property. This complements a previous result by R. Meyer and R. Nest. For this I have developed a reformulation of the equivariant Kasparov in terms of explicit formulae using the representation theory of a given compact quantum group. This reformulation turns out to be a key technical tool to address the general construction of compact bicrossed products in relation with approximation properties, which is now part of my future research project.
On a different note, I have defined a quantum equivariant Künneth formula using the notion of Künneth functor in order to relate the quantum BC property to the Künneth formula.
Due to their complexity, some problems have not been solved as expected. For instance, this is is the case for the problems addressed around CEP. More time and further investigation is needed. I have studied the recently discovered connections between compact quantum groups and non-local games in the framework of quantum information theory in order to address relevant questions around CEP and quantum strategies for the graph isomorphism game. In the course of the action, however, I have acquired new directions to address these problems by discovering supplementary open questions related to my original objective.
All in all, the research carried out during the project has led to significant steps towards advancing the development of the BC property in the quantum setting. Since all my results are freely available on the Internet, the scientific papers produced during the fellowship will allow to promote further investigations in the subject, which is already the case. In addition, the exploration of different directions to develop the proposed project has allowed me to configure a new future research plan with supplementary and innovative approaches to the subject.