标题： Kaplansky—Hilbert模和具有离散谱的扩张中的预紧性概念 显示英文标题

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

摘要： 本文是我们关于经典Furstenberg-Zimmer理论函数分析核心工作的延续。 我们在格序空间框架下引入并研究了全序有界性和一致全序有界性的概念。 两者之一推广了度量空间理论中已知的普通预紧性概念。 这些新概念随后被用来定义和刻画一般测度保持系统的“紧扩张”（对底概率空间和作用群没有限制）。 特别是，我们（重新）证明了紧扩张与具有离散谱的扩张是一回事。 最后，我们证明，在自然假设下，Kaplansky-Banach模的一个子集是全序有界的当且仅当它是循环紧的（按照Kusraev的意义）。 显示英文摘要

摘要： 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).