Topological invariance of the homological index

Abstract

R. W. Carey and J. Pincus in [6] proposed an index theory for non-Fredholm bounded operators T on a separable Hilbert space H such that TT∗−T∗T is in the trace class. We showed in [3] using Dirac-type operators acting on sections of bundles over R2n that we could construct bounded operators T satisfying the more general condition that the operator (1−TT∗)n−(1−T∗T)n is in the trace class. We proposed there a ‘homological index’ for these Dirac-type operators given by Tr((1−TT∗)n−(1−T∗T)n). In this paper we show that the index introduced in [3] represents the result of a paring between a cyclic homology theory for the algebra generated by T and T∗ and its dual cohomology theory. This leads us to establish the homotopy invariance of our homological index (in the sense of cyclic theory). We are then able to define in a very general fashion a homological index for certain unbounded operators and prove invariance of this index under a class of unbounded perturbations.