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