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 and manifolds, we get a topological isomorphism between the cohomology groups of an A-elliptic complex and the space of harmonic elements of the complex.
Consequently, the cohomology groups appear to be finitely generated projective C*-Hilbert modules and especially, Banach spaces. We also prove that in the category of Hilbert A-modules and continuous adjointable Hilbert A-module homomorphisms, the property of a complex of being self-adjoint parametrix possessing characterizes the complexes of Hodge type. (C) 2016 Published by Elsevier B.V.