Fredholm complexes of Hilbert C*-modules

TL;DR

This paper studies Fredholm complexes of Hilbert C*-modules, providing characterizations, defining an index in K-theory, and analyzing its stability and alternative expressions under certain decompositions.

Contribution

It introduces new characterizations of the Fredholm property and defines a K-theory valued index for complexes of Hilbert C*-modules.

Findings

Fredholm property characterized in multiple equivalent ways

Defined a K-theory index for these complexes

Proved stability of the index under perturbations

Abstract

We investigate complexes of Hilbert C*-modules, which are cochain complexes with (unbounded) regular operators between Hilbert C*-modules as differential maps. In particular, we provide various equivalent characterizations of the Fredholm property for such complexes of Hilbert C*-modules, and we define the Fredholm index taking values in the K-theory group of the C*-algebra. Among other properties of this index, we prove the stability under small or relatively compact perturbations, and we obtain alternative expressions for the index under the existence of a (weak or strong) Hodge decomposition.