Precompactness notions in Kaplansky--Hilbert modules and extensions with discrete spectrum
Markus Haase, Henrik Kreidler
TL;DR
The paper develops a purely functional-analytic framework to extend Furstenberg–Zimmer-type structure theorems beyond standard spaces by introducing total order-boundedness notions in lattice-normed spaces over a Stone algebra and relating them to KH-dynamical systems. It proves that every bounded subset of a finite-rank Kaplansky–Hilbert module is uniformly totally order-bounded and shows an equivalence between Tao’s zonotope-based compactness and uniform total order-boundedness. It then characterizes compact extensions as precisely those with discrete spectrum in the abstract KH-framework, linking conditional almost periodicity to precompactness via order-analytic notions. Finally, it relates total order-boundedness to Kusraev’s relative cyclical compactness, giving a mix-closure criterion in Kaplansky–Banach modules. Together, the results provide a cohesive, abstract functional-analytic path to classical ergodic-theoretic dichotomies without regularity assumptions on the underlying spaces or groups.
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).