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

TL;DR

This paper extends the theory of measure-preserving systems by introducing new notions of total order-boundedness in lattice-ordered spaces, characterizing compact extensions and discrete spectrum in a general setting, and linking order-boundedness with cyclic compactness.

Contribution

It introduces total order-boundedness concepts in lattice-ordered spaces and characterizes compact extensions and discrete spectrum without restrictions on spaces or groups.

Findings

Compact extensions are equivalent to extensions with discrete spectrum.

Total order-boundedness characterizes cyclically compact subsets.

New notions generalize classical precompactness to measure-preserving systems.

Abstract

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 nor 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).