Hodge theory for elliptic complexes over unital $C^*$-algebras

TL;DR

This paper develops a Hodge theory for elliptic complexes of pseudodifferential operators acting on sections of $A$-Hilbert bundles over manifolds, where $A$ is a $C^*$-algebra, showing cohomology groups are finitely generated $A$-modules.

Contribution

It introduces a notion of ellipticity for complexes over $C^*$-algebra bundles and proves the cohomology groups are finitely generated $A$-modules under certain conditions.

Findings

Cohomology groups are norm complete finitely generated $A$-modules.

Establishes a Hodge theory for $A$-elliptic complexes.

Shows the importance of the closedness of Laplacian images.

Abstract

We introduce a notion of ellipticity of complexes of linear pseudodifferential operators acting on sections of $A$-Hilbert bundles over smooth manifolds, $A$ being a $C^*$-algebra. We prove that the cohomology groups of an $A$-elliptic pseudodifferential complex in finitely generated projective $A$-Hilbert bundles over a compact manifold are norm complete finitely generated $A$-modules if the images of the associated Laplacians are closed. This establishes a Hodge theory for these structures.