K-theory, C*-algebras and Index Theory of Elliptic Operators

Indiana University, Bloomington IN

Investigators

Abstract

Abstract Kaminker The researcher will work on three projects. Investigation will be continued on the connection between hyperbolic dynamical systems and K-theory of C*-algebras. In particular, the zeta function of a Smale space will be expressed in terms of the K-theory of the stable algebra and the induced homomorphisms. This leads to a general program to use the K-theory of these C*-algebras as a replacement for ordinary homology groups which were used in the past to study hyperbolic systems. The second direction involves the study of the relation between the Exactness conjecture and the Novikov conjecture for finitely presented groups. If a group acts amenably on a compact space then both of these conjectures hold. The relation between the existence of such an action and approximation properties for the reduced C*-algebra will be studied and precise relations between the two conjectures will be established. In a third direction, work will be done on obtaining a higher index formula for regular Lie groupoids. This will provide a formula for the pairing of certain cyclic cocycle on the smooth convolution algebra of the groupoid with the class in K-theory representing the analytic index of an elliptic operator. These projects are part of a general program to see how symmetry, in its various forms, effects different parts of mathematics. It is often the case that unexpected symmetry has important implications. Solutions to important equations in physics and engineering have been discovered and studied by using the fact that their form must be unchanged under certain transformations. It is hoped that the results of the present project will have similar applications.