L2-index formula for proper cocompact group actions

Abstract

Indices are analytical invariants to some elliptic operators and an index formula provides a way to interpret the analysis quantity using the topological invariants. The thesis computes the L2-index of a properly supported elliptic pseudo-differential operator which acts on a complete Riemannian manifold and being invariant under a properly cocompact group action. The group is assumed to be a locally compact one admitting an invariant Haar measure. The L2-index of an invariant elliptic operator is defined by taking the von Neumann trace of the higher index in the K-theory of the group C*-algebra. The thesis provides a cohomological formula for the L2-index for elliptic operators with properly cocompact group actions using the KK-theory and the heat kernel method. The formula is a generalization to the Atiyah's L2-index theorem for free cocompact group actions and to the Connes and Moscovici's L2-index formula for homogenous space of unimodular Lie group.