  For a symplectic manifold admitting a metaplectic structure and for a Kuiper
map, we construct a complex of differential operators acting on exterior
differential forms with values in the dual of the Kostant's symplectic spinor
bundle. Defining a Hilbert $C^*$-structure on this bundle for a suitable
$C^*$-algebra, we obtain an elliptic $C^*$-complex in the sense of
Mishchenko--Fomenko. Its cohomology groups appear to be finitely generated
projective Hilbert $C^*$-modules. The paper can serve as a guide for handling
of differential complexes and PDEs on Hilbert bundles
