WSU

Department of Mathematics
Topology Seminar 2006-2007

Wayne State University
College of Science


All talks are held Tuesdays at 2:00 PM in 1285 FAB, unless otherwise noted.
Contact Dan Isaksen (isaksen AT math.wayne.edu) for more information

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Colloquium
Mathematics Department home page



Date: October 2, 2007

Speaker: Chris Phillips, University of Oregon

Title: Crossed products of irrational rotation algebras by finite groups

Abstract:   For each theta in R, the rotation algebra A_theta is defined to be the universal C*-algebra generated by two unitary operators u and v satisfying v u = e^{2 pi i theta} u v.  If theta = 0, this C*-algebra is just C (S^1 \times S^1), the algebra of continuous complex valued functions on the torus.  Hence, for general theta this C*-algebra is sometimes called a noncommutative torus.

There is an action of SL_2 (Z) on A_theta which generalizes the action of SL_2 (Z) on S^1 \times S^1 obtained via the identification S^1 \times S^1 = R^2 / Z^2.  In particular, the subgroups of SL_2 (Z) isomorphic to Z / 2Z, Z / 3Z, Z / 4Z, and Z / 6Z all act on A_theta.  For example, the action of Z / 4Z is generated by the automorphism which sends u to v and v to u^*.

For rational theta, the (topological) K-theory of the crossed products of A_theta by these groups has been calculated.  The results suggest the conjecture that, for theta irrational, the crossed products are AF-algebras, that is, direct limits of finite dimensional C*-algebras.  This has been known for some time for Z/2Z. We prove this conjecture for the other three groups.  One might hope to be able to write down an approximating sequence of finite dimensional subalgebras.  In fact, the proof uses the Baum-Connes Conjecture to compute the K-theory of the crossed products, and the Elliott classification program to deduce their isomorphism types.  These are two central areas of research in C*-algebras which previously have had little contact.

In this talk, I will describe the Baum-Connes Conjecture and the Elliott classification program, and role that they play in the proof.  (I will say more about the Elliott program than in the colloquium talk the previous day.)  There will be little technical detail.

This is joint work with Siegfried Echterhoff, Wolfgang Lueck, and Sam Walters.