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For a unital Banach C * -algebra A, we prove that the cohomology groups of A-elliptic complexes of pseudodifferential operators in finitely generated projective A-Hilbert bundles over compact manifolds are norm complete topological vector spaces and finitely generated A-modules provided the images of certain extensions of the so called associated Laplacians are closed. This establishes a Hodge type theory for these structures.

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 finitely generated projective C * -Hilbert modules and especially, 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.Math. Subj. Class. 2010: Primary 46M18; Secondary 46L08, 46M20

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