Hodge theory for complexes over C*-algebras with an application to A-ellipticity

Abstract

For a class of co-chain complexes in the category of pre-Hilbert -modules, we prove that their cohomology groups equipped with the canonical quotient topology are pre-Hilbert -modules, and derive the Hodge theory and, in particular, the Hodge decomposition for them. As an application, we show that -elliptic complexes of pseudodifferential operators acting on sections of finitely generated projective -Hilbert bundles over compact manifolds belong to this class if the images of the continuous extensions of their associated Laplace operators are closed.

Moreover, we prove that the cohomology groups of these complexes share the structure of the fibers, in the sense that they are also finitely generated projective Hilbert A-modules.

Keywords

• Hodge theory
• Hilbert C*-modules
• C*-Hilbert bundles
• Elliptic systems of partial differential equations