Elliptic complexes over $C^*$-algebras of compact operators

TL;DR

This paper establishes Hodge theory for elliptic complexes over $C^*$-algebras of compact operators, showing cohomology groups are Banach spaces and linking them to harmonic elements.

Contribution

It proves Hodge theory holds for $A$-elliptic complexes over compact operator $C^*$-algebras, including a topological isomorphism with harmonic elements.

Findings

Cohomology groups are Banach spaces.

Hodge theory holds for $A$-elliptic complexes.

Cohomology is isomorphic to harmonic elements.

Abstract

For a $C^*$-algebra $A$ of compact operators and a compact manifold $M,$ we prove that the Hodge theory holds for $A$-elliptic complexes of pseudodifferential operators acting on smooth sections of finitely generated projective $A$-Hilbert bundles over $M.$ For these $C^*$-algebras, we get also a topological isomorphism between the cohomology groups of an $A$-elliptic complex and the space of harmonic elements. Consequently, the cohomology groups appear to be Banach spaces. We prove as well, that if the Hodge theory holds for a complex in the category of Hilbert $A$-modules and continuous adjointable Hilbert $A$-module homomorphisms, the complex is self-adjoint parametrix possessing.