AM -Spaces from a Locally Solid Vector Lattice Point of View with Applications
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AM-Spaces from a Locally Solid Vector Lattice Point of View with Applications Omid Zabeti1 Received: 27 March 2020 / Revised: 15 August 2020 / Accepted: 27 August 2020 © Iranian Mathematical Society 2020
Abstract Suppose X is a locally solid vector lattice. In this paper, we introduce the notion of “AM-property” in X as an extension of the property fulfilled by 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 possess the Lebesgue or Levi property if and only if so is X . Keywords Locally solid vector lattice · Bounded operator · AM-property · Levi property · Lebesgue property Mathematics Subject Classification 46A40 · 47B65 · 46A32
1 Motivation and Preliminaries Let us start with some motivation. Suppose K is a compact Hausdorff topological space and C(K ) is the Banach lattice of all real-valued continuous functions on K . In general, C(K )-spaces have important properties in the category of all Banach lattices; for example in a C(K )-space, the Levi property and order completeness agree, order boundedness and boundedness coincide and so on. In addition, every AM-space can be isometrically embedded into a C(K )-space (the remarkable Kakutani theorem); recall that a Banach lattice E which satisfies x ∨ y = x ∨ y for every x, y ∈ E + is
Communicated by Fereshteh Sady.
B 1
Omid Zabeti [email protected] Department of Mathematics, University of Sistan and Baluchestan, P.O. Box: 98135-674, Zahedan, Iran
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Bulletin of the Iranian Mathematical Society
termed as an AM-space. On the other hand, an interesting question regarding operators between normed lattices is the following. Is there any relation between continuous operators and order bounded ones. It is known that when the domain is a Banach lattice, every order bounded operator is continuous. In general, the answer for the other direction is almost negative. Nevertheless, in some special cases, this can happen; for C(K )-spaces E and F, an operator T : E → F is continuous if and only if it is order bounded; note that we cannot replace C(K )-space with an AM-space (see [2, Example 4.73]). Furthermore, in a locally solid vector lattice, there are several non-equivalent ways to define bounded operators. So, it is interesting to investigate the relations between these types of bounded operators and order bounded ones. This is what our paper is about. Moreover, it is shown in [3] that, under some mild assumptions, each class of bounded order bounded operators between locally solid vector lattices has a lattice structure, too. So, another interesting direction is to find situations under which a known property regarding the lattice structure such as the Levi or the Lebesgue property can be transformed between the space of bound
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