Hodge theory for elliptic complexes over unital C^*-algebras

For a unital $C^{*}$ -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 finitely generated $A$ -modules and Banach spaces provided the images of certain extensions of the so-called associated Laplacians are closed. We also prove that under this condition, the cohomology groups are isomorphic to the kernels of the associated Laplacians. This establishes a Hodge theory for these structures.