Oberwolfach Seminar: Topological K-Theory for Noncommutative Algebras and Applications

Date:

May 15th - May 21st, 2005

Organizers:

Joachim Cuntz, Münster

Ralf Meyer, Münster

Jonathan Rosenberg, Maryland

Subjects:

Many different situations, arising from subjects ranging from mathematical physics and differential geometry to number theory, can be described efficiently by noncommutative algebras. Examples include algebras of differential or pseudodifferential operators, group algebras and foliation algebras, algebras of intertwining operators for representations, algebras of observables in quantum physics, and Hecke algebras. The new field of noncommutative geometry applies ideas from geometry, but with the algebra of functions on a space replaced by such a noncommutative algebra.

The first aim of the seminar is to introduce the ideas of topological K-theory for noncommutative algebras. We first plan to study the ordinary (monovariant) K-theory for Banach and C*-algebras. We will introduce bivariant K-theories through the example of the recently developed bivariant K-theory for locally convex algebras. Other bivariant K-theories (such as in particular the ``classical'' Kasparov theory on the category of C*-algebras) have exactly analogous properties, but require more sophisticated techniques for their definition.

A second aim is the discussion of some typical applications of these techniques. We plan a discussion of bivariant versions of the Atiyah-Singer index theorem. The most general version of this theorem determines the K-homology class determined by the extension defined by the algebra of pseudodifferential operators on a compact manifold. It contains significantly more information than the classical index theorem. Another topic that we plan to discuss is the so-called ``twisted K-theory'' that has received much attention recently among mathematical physicists, and which has a very natural interpretation using the K-theory of certain noncommutative C*-algebras.

Prerequisites:

Manifolds and vector bundles, basics on K⁰ of a space and the definition of K₀ for a ring, basics on pseudodifferential operators and linear elliptic PDEs, basics of functional analysis.

Literature:

1. P. Baum, R. Douglas, and M. Taylor, Cycles and relative cycles in analytic K-homology, J. Differential Geom. 30 (1989), no. 3, 761-804.
2. B. Blackadar, K-theory for operator algebras, 2nd ed., Math. Sci. Res. Inst. Publ., 5, Cambridge Univ. Press, Cambridge, 1998.
3. A. Connes, Noncommutative geometry, Academic Press, San Diego, 1994.
4. J. Cuntz, Bivariante K-Theorie für lokalkonvexe Algebren und der Chern-Connes-Charakter, Doc. Math. 2 (1997), 139-182.
5. J. Cuntz, G. Skandalis, and B. Tsygan, Cyclic homology in non-commutative geometry, Encyclopaedia of Mathematical Sciences, 121: Operator Algebras and Non-commutative Geometry, II, Springer-Verlag, Berlin, 2004.
6. M. Karoubi, K-theory: An introduction, Grundlehren der Math. Wissenschaften, 226, Springer-Verlag, Berlin-New York, 1978.
7. G. Kasparov, The operator K-functor and extensions of C*-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 3, 571-636, 719. English translation: Math. USSR-Izv. 16 (1981), 513-572.
8. G. Kasparov, K-theory, group C*-algebras, and higher signatures (conspectus), in Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993), London Math. Soc. Lecture Notes, 226, Cambridge Univ. Press, Cambridge, 1995, pp. 101-146.
9. I. Raeburn and D. Williams, Morita equivalence and continuous-trace C*-algebras, Math. Surveys and Monographs, 60, Amer. Math. Soc., Providence, 1998.
10. J. Rosenberg, Algebraic K-theory and its applications, Graduate Texts in Math., 147, Springer-Verlag, New York, 1994.

Deadline for applications:

April 1st, 2005

The seminars take place at the Mathematisches Forschungsinstitut Oberwolfach. The number of participants is restricted to 24. The Institute covers accommodation and food. Travel expenses cannot be reimbursed. Applications including

• full name and address, including e-mail address
• present position, university
• name of supervisor of Ph.D. thesis
• a short summary of previous work and interest

should be sent as hard copy or by e-mail (.ps or .pdf file) to:

Prof. Dr. Gert-Martin Greuel
Universität Kaiserslautern
Fachbereich Mathematik
Erwin Schrödingerstr.
67663 Kaiserslautern, Germany
.