# Project C3: (Semi-)group C*-algebras and their invariants

## Principal investigator

## Participating scientists

- Prof. Dr. Arthur Bartels
- Christian Boenicke
- Sayan Chakraborty
- Prof. Dr. Dr. h.c. Joachim Cuntz
- Andrea Kasprowski
- Kang Li
- Hamed Nikpey
- Dr. Walther Paravicini
- Dr. Ansgar Schneider
- Dr. Jan Spakula
- Dr. Christian Voigt
- Dr. Stefan Wagner

## Summary

The most fundamental construction in the intersection of Harmonic Analysis, Topology and Operator Algebras is given by the convolution algebra L^{1}(G) of a locally compact group G and its full and reduced C*-completions C*(G) and C*

_{r}(G). In this project we use the Baum-Connes conjecture as a tool for the study of the topological structure of these algebras which, in the spirit of noncommutative geometry/topology, represent the unitary representations of the underlying group G.

In general, the space Ĝ of equivalence classes of irreducible representations of a nonabelian locally compact group G is a very badly behaved topological space. Although it carries a natural topology, called the "Jacobson topology", this topology very often fails any of the usual separation axioms - indeed, for discrete groups they even fail the T

_{0}-axiom unless the group is virtually abelian. However, in the sense of non-commutative Geometry/Topology we may regard C*(G) (resp. C*

_{r}(G)) as generalized (non-commutative) function spaces on Ĝ (resp. the tempered dual Ĝ

_{λ}), which have a very rich structure, and which allow to use non-commutative analogues of classical methods from Algebraic Topology (like K-theory, K-homology, cyclic Homology and others) for a more detailed study of the given structures. Aside of the classical group algebras, it is also useful to study group algebras which are "twisted" by some circle valued 2-cocycle on G. In Harmonic Analysis these algebras appear in the study of projective representations of G. On the other hand, many important examples of "non-commutative geometric spaces" appear as such algebras - the most prominent examples are given by the non-commutative n-tori, which are just the twisted group algebras of Z

^{n}. Other important generalizations studied in this project are the locally compact quantum groups as introduced by Kustermans and Vaes.

The Baum-Connes conjecture, which in its original form describes the K-theory of C*

_{r}(G) in terms of the K-homology of a certain classifying space of G, provides a major tool for the study of the topological structure of C*

_{r}(G). Variants of this conjecture also exist for the other group algebras discussed above. The study of this conjecture and its relation to the geometry/topology and the representation theory of G will form a central rôle in this project.

In the following, we give a short list of some particular topics we plan to investigate within our project.

- Explicit computations of K-theory groups for interesting (twisted) group-algebras.
- Lefschetz fixed-point theorems and Poincaré duality for (twisted) group algebras and for certain discrete quantum groups.
- Extend methods developed for the study of isomorphism conjectures in Topology (like the Farrel-Jones conjecture for the Ltheory of group rings) to the study of the Baum-Connes conjecture.
- Establish further permanence results for the Bost-conjecture (the
Banach-algebra analogue of the Baum-Connes conjecture) for the
K-theory of L
^{1}(G). - Give a proof on the Baum-Connes conjecture for linear algebraic groups over local fields with positive characteristic.
- Study the connection between K-theory of C*
_{r}(G) and the representation theory of G for almost connected Lie-groups. - Prove an analogue of the Baum-Connes conjecture for certain discrete quantum groups.