Thomas Schick: L2-index theorems, KK-theory, and connections

Let M be a compact manifold. and D a Dirac type differential operator on M. Let A be a C^*-algebra. Given a bundle of A-modules over M (with connection), the operator D can be twisted with this bundle. One can then use a trace on A to define numerical indices of this twisted operator. We prove an explicit formula for this index. Our result does complement the Mishchenko-Fomenko index theorem valid in the same situation. As a corollary, we see that these numerical indices don't give additional information if the twisting bundle is flat. There are different versions of the indices which can be obtained. An important part of the paper is to give complete proofs that they coincide. In particular, we reprove Atiyah's L2-index theorem, and we establish the (well known but not well documented) equality of Atiyah's definition of the L2-index with a K-theoretic definition. Some of our calculations are done in the framework of bivariant KK-theory. MSC 2000: 19K35, 19K56, 46M20, 46L80, 58J22 schick@umath.uni-goettingen.de.

Thomas Schick (schick@math.uni-goettingen.de)