A partial order on subsets of Baer bimodules with applications to C∗-modules

Abstract

In this paper, we introduce the concept of Baer [Formula: see text]-sets. Using this notion, we define Rickart, Baer, quasi-Baer and [Formula: see text]-Baer [Formula: see text]-bimodules, respectively. We show how these conditions relate to each other. We also develop new properties of the minus binary relation, [Formula: see text]-, we extend the relation [Formula: see text]- to [Formula: see text]-bimodules and use it to characterize the aforementioned Rickart, Baer, quasi-Baer, and [Formula: see text]-Baer [Formula: see text]-bimodules. Moreover, we specify subsets [Formula: see text] of the power set of a [Formula: see text]-bimodule for which [Formula: see text]- determines a partial order and for which [Formula: see text]- is a lattice. We analyze the relation [Formula: see text]- by examining the associated Baer [Formula: see text]-sets. Finally, we apply our results to [Formula: see text]-modules. Examples are provided to illustrate and delimit our results.