Precompactness notions in Kaplansky–Hilbert modules and extensions with discrete spectrum

This paper is a continuation of our work on the functional-analytic core of the classical Furstenberg–Zimmer theory. We introduce and study (in the framework of lattice-ordered spaces) the notions of total order-boundedness and uniform total order-boundedness. Either one generalizes the concept of ordinary precompactness known from metric space theory. These new notions are then used to define and characterize “compact extensions” of general measure-preserving systems (with no restrictions on the underlying probability spaces or on the acting groups). In particular, it is (re)proved that compact extensions and extensions with discrete spectrum are one and the same thing. Finally, we show that under natural hypotheses a subset of a Kaplansky–Banach module is totally order-bounded if and only if it is cyclically compact (in the sense of Kusraev).