$AM$-spaces from a locally solid vector lattice point of view with applications

TL;DR

This paper introduces the $AM$-property in locally solid vector lattices, characterizes spaces where bounded and order bounded sets coincide, and explores conditions for bounded operators to have Lebesgue or Levi properties.

Contribution

It extends the concept of $AM$-spaces to locally solid vector lattices and provides characterizations linking bounded sets, operators, and properties like Lebesgue and Levi.

Findings

Spaces where bounded and order bounded sets agree are characterized.

Conditions for bounded operators to be order bounded are established.

Equivalence of Lebesgue or Levi properties between operators and the space is shown.

Abstract

Suppose $X$ is a locally solid vector lattice. In this paper, we introduce the notion "$AM$-property" in $X$ as an extension for $AM$-spaces in the category of all Banach lattices. With the aid of this concept, we characterize spaces in which bounded sets and order bounded sets agree. This, in turn, characterizes conditions under which each class of bounded operators on $X$ is order bounded and vice versa. Also, we show that under some natural assumptions, different types of bounded order bounded operators on $X$ have the Lebesgue or Levi property if and only if so is $X$.