  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 $C^*$-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.
