What question did this study set out to answer?

The research aims to characterize Fredholm properties in complexes of Hilbert C*-modules and define the Fredholm index.

December 21, 2025Open Access

Fredholm complexes of Hilbert C*-modules

BVBrian Villegas-VillalpandoAutonomous University of Aguascalientes KDKoen van den DungenUniversity of Bonn

Key Points

The research aims to characterize Fredholm properties in complexes of Hilbert C*-modules and define the Fredholm index.
Investigated cochain complexes with unbounded regular operators
Defined Fredholm index in the context of K-theory
Proved stability under small and relatively compact perturbations
Explored alternative expressions of the index with Hodge decomposition
Provided equivalent characterizations of the Fredholm property
Demonstrated stability of the Fredholm index under perturbations
Established alternative expressions for the Fredholm index involving Hodge decomposition

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.

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