 

Thomas Schick
L^{2}index theorems, KKtheory, and connections


Published: 
September 5, 2005

Keywords: 
Ktheory, index theory,curvature, Chern character, C^{*}algebra, L^{2}index, MishchenkoFomenko index theorem, trace 
Subject: 
19K35, 19K56, 46M20, 46L80, 58J22 


Abstract
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
Amodules 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 MishchenkoFomenko 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 L^{2}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
coincide. A
substantial part of the paper is a complete proof of their equality. In
particular, we establish the (wellknown but not welldocumented)
equality of Atiyah's definition of the L^{2}index with a Ktheoretic 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 Ktheoretic information about
the index of the twisted operator.
Some of our calculations are done in the framework of bivariant
KKtheory.


Author information
FB Mathematik, Universität Göttingen, Bunsenstr. 3, 37073 Göttingen, Germany
schick@unimath.gwdg.de
http://www.unimath.gwdg.de/schick

