L2-index theorems, KK-theory, and connections
||September 5, 2005
||K-theory, index theory,curvature, Chern character, C*-algebra, L2-index, Mishchenko-Fomenko index theorem, trace
||19K35, 19K56, 46M20, 46L80, 58J22
Let M be a compact manifold and D a Dirac type differential
operator on M. Let A be a C*-algebra. Given a bundle W (with connection) of
A-modules over M,
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 these indices. Our result does complement
the Mishchenko-Fomenko index theorem valid in the same situation.
We establish generalizations of these explicit index formulas if the trace is only
defined on a dense and holomorphically closed subalgebra B.
As a corollary, we prove a generalized Atiyah L2-index theorem if
the twisting bundle is flat.
There are actually many different ways to define these numerical
indices. From their construction, it is not clear at all that they
substantial part of the paper is a complete proof of their equality. In
particular, we establish the (well-known but not well-documented)
equality of Atiyah's definition of the L2-index with a K-theoretic definition.
In case A is a von Neumann algebra of type 2, we put special
emphasis on the calculation and interpretation of the center valued
index. This completely contains all the K-theoretic information about
the index of the twisted operator.
Some of our calculations are done in the framework of bivariant
FB Mathematik, Universität Göttingen, Bunsenstr. 3, 37073 Göttingen, Germany