The K-Theory and Symmetry of Geometry Given C*-Algebras

Research group:
The K-Theory and Symmetry of Geometry Given C*-Algebras

16 January, 2012, Warsaw

Organizer

  • Piotr M. Hajac

Programme

10:15, lecture room 322

THE K-THEORY OF FREE QUANTUM GROUPS, Christian Voigt (Universitat Munster, Germany)

This talk concerns the K-theory of free quantum groups in the sense of Wang and Van Daele. More precisely, we show that the free products of free unitary and free orthogonal quantum groups are K-amenable, and establish an analogue of the Pimser-Voiculescu exact sequence. As a particular consequence, we obtain an explicit computation of the K-theory of free quantum groups. Our approach relies on a generalization of Baum-Connes conjecture methods to the framework of discrete quantum groups. It is based on the categorical reformulation of the Baum-Connes conjecture developed by Meyer and Nest. As a main result, we show that the gamma-element of any free quantum groups equals 1. As an important ingredient in the proof, we adapt the Dirac-dual-Dirac method for groups acting on trees to the quantum case. We use this to extend some permanence properties of the Baum-Connes conjecture to our setting (Joint work with Roland Vergnioux).

14:15, lecture room 322

INVARIANTS FOR A CONFORMALLY REGULAR PENTAGONAL TILING OF THE PLANE,
Maria Ramirez-Solano (Kobenhavns Universitet, Denmark)

The Bowers and Stephenson conformally regular pentagonal tiling of the plane enjoys remarkable combinatorial and geometric properties. Since it does not have finite local complexity in any usual sense, it is beyond the standard tiling theory. On the other hand, the tiling can be completely described by its combinatorial data that, rather automatically, has finite local complexity. With the aim to compute its K-theory, we construct the hull and C*-algebra of this tiling solely from its combinatorial data. As the tiling possesses no natural R² action by translation, there is no a priori reason to expect that the K-theory of the C*-algebra of the tiling is the same as the K-theory or cohomology of the hull of the tiling, and it would be very interesting if they were different.

Participants

  • Piotr M. Hajac (IMPAN/University of Warsaw)
  • Paweł Kasprzak (IMPAN/ University of Warsaw)
  • Tomasz Maszczyk (IMPAN/ University of Warsaw)
  • Maria Ramirez-Solano (Kobenhavns Universitet, Denmark)
  • Jan Rudnik (IMPAN)
  • Andrzej Sitarz (IMPAN/ Jagiellonian University)
  • Christian Voigt (Universitat Munster, Germany)