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



Principal investigator

Participating scientists

Summary

The most fundamental construction in the intersection of Harmonic Analysis, Topology and Operator Algebras is given by the convolution algebra L1(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 T0-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 Ĝλ), 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 Zn. 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 L1(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.

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