Hodge decomposition for elliptic complexes over unital $C^{\ast }$-algebras

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.

13 pages, submitted Banach Journal of Mathematical Analysis

References in corpus (2)

• Spin Foams and Noncommutative Geometry
• Hodge theory for elliptic complexes over unital $C^{\ast }$-algebras

Cited by in corpus (3)

• On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for $C^{\ast }$-Dynamical Systems
• Elliptic complexes over $C^{\ast }$-algebras of compact operators
• Induced $C^{\ast }$-complexes in metaplectic geometry