  In this paper, a decomposition theorem for (covariant) unitary group
representations on Kaplansky-Hilbert modules over Stone algebras is
established, which generalizes the well-known Hilbert space case (where it
coincides with the decomposition of Jacobs, de Leeuw and Glicksberg). The proof
rests heavily on the operator theory on Kaplansky-Hilbert modules, in
particular the spectral theorem for Hilbert-Schmidt homomorphisms on such
modules. As an application, a generalization of the celebrated
Furstenberg-Zimmer structure theorem to the case of measure-preserving actions
of arbitrary groups on arbitrary probability spaces is established.
