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

We introduce a notion of ellipticity of complexes of linear pseudodifferential operators acting on sections of $A$-Hilbert bundles over smooth manifolds, $A$ being a Banach $C^*$-algebra. We prove that the cohomology groups of an $A$-elliptic pseudodifferential complex in finitely generated projective $A$-Hilbert bundles over a compact manifold are norm complete finitely generated $A$-modules if the images of the associated Laplacians are closed. This establishes a Hodge theory for these structures.