Elliptic complexes over C∗C∗-algebras of compact operators

For a C∗C∗-algebra AA of compact operators and a compact manifold MM, we prove that the Hodge theory holds for AA-elliptic complexes of pseudodifferential operators acting on smooth sections of finitely generated projective AA-Hilbert bundles over MM. For these C∗C∗-algebras and manifolds, we get a topological isomorphism between the cohomology groups of an AA-elliptic complex and the space of harmonic elements of the complex. Consequently, the cohomology groups appear to be finitely generated projective C∗C∗-Hilbert modules and especially, Banach spaces. We also prove that in the category of Hilbert AA-modules and continuous adjointable Hilbert AA-module homomorphisms, the property of a complex of being self-adjoint parametrix possessing characterizes the complexes of Hodge type.