Topological invariance of the homological index

Abstract

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 {\mathcal{H}} such that T ⁢ T * - T * ⁢ T {TT^{*}-T^{*}T} is in the trace class. We showed in [3] using Dirac-type operators acting on sections of bundles over ℝ 2 ⁢ n {\mathbb{R}^{2n}} that we could construct bounded operators T satisfying the more general condition that the operator ( 1 - T ⁢ T * ) n - ( 1 - T * ⁢ T ) n {(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 - T ⁢ T * ) n - ( 1 - T * ⁢ T ) n ) {{\rm 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 * {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.