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

For a certain class of complexes of pre-Hilbert $A$-modules, we prove that their cohomology groups equipped with a canonical quotient structure are again pre-Hilbert $A$-modules and derive the Hodge decomposition for them. We call these complexes self-adjoint parametrix possessing. We show that $A$-elliptic complexes of differential operator acting on sections of finitely generated projective $A$-Hilbert bundles over compact manifolds have this property if the images of certain extensions of their Laplacians are closed.