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

TL;DR

This paper extends Hodge theory to elliptic complexes over unital $C^*$-algebras, establishing a decomposition for cohomology groups of certain self-adjoint complexes of pre-Hilbert modules.

Contribution

It introduces the class of self-adjoint parametrix possessing complexes and proves Hodge decomposition for them, generalizing classical results to $C^*$-algebra settings.

Findings

Cohomology groups are pre-Hilbert $A$-modules with canonical quotient structures.

Hodge decomposition holds for these complexes under certain closed image conditions.

Elliptic complexes of differential operators on $A$-Hilbert bundles satisfy these conditions on compact manifolds.

Abstract

For a certain class of complexes of pre-Hilbert $A$-modules, we prove that their cohomology groups equipped with a canonical quotient structure are again pre-Hilbert $A$-modules and derive the Hodge decomposition for them. We call these complexes self-adjoint parametrix possessing. We show that $A$-elliptic complexes of differential operator acting on sections of finitely generated projective $A$-Hilbert bundles over compact manifolds have this property if the images of certain extensions of their Laplacians are closed.