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A topology on the Fremlin tensor product between locally solid vector lattices with applications

Omid Zabeti

TL;DR

The paper addresses the problem of equipping the Fremlin tensor product $E\overline{\otimes}F$ with a locally solid topology and determining when the canonical bilinear map is uniformly continuous under unbounded convergences. It develops a locally solid topology $τ_{E\overline{\otimes}F}$ from solid zero neighborhoods and proves density of $E\otimes F$, then demonstrates that the Fremlin projective tensor product preserves both $un$-convergence and $uaw$-convergence for Banach lattices, initially under mild assumptions and ultimately in full generality using a key tensor-product inequality. A central contribution is a topological framework that yields preservation results without extra hypotheses and provides a simpler proof of a main result in the literature (Z:25). Overall, the work enhances the understanding of unbounded convergence in tensor products of vector lattices and offers a robust method with potential applications in analysis on Banach lattices and locally solid vector lattices.

Abstract

Let $E$ and $F$ be locally solid vector lattices. We introduce a locally solid topology on the Fremlin tensor product $E\overline{\otimes}F$ and we denote it by $τ_{E\overline{\otimes}F}$. It extends the Fremlin projective tensor product in the setting of Banach lattices. The main purpose of the paper is to determine conditions under which, the canonical bilinear mapping \[E\times F\to (E\overline{\otimes}F,τ_{E\overline{\otimes}F}),\hspace{0.50cm} (x,y)\mapsto x\otimes y,\] is uniformly continuous when $E$ and $F$ are equipped with appropriate unbounded convergences whose associated topologies make the corresponding spaces locally solid vector lattices. By means of an inequality for tensor products in vector lattices and by exploiting the structure of the topology $τ_{E\overline{\otimes}F}$, we show, in particular, that the Fremlin tensor product between Banach lattices preserves several important classes of unbounded convergences.

A topology on the Fremlin tensor product between locally solid vector lattices with applications

TL;DR

The paper addresses the problem of equipping the Fremlin tensor product with a locally solid topology and determining when the canonical bilinear map is uniformly continuous under unbounded convergences. It develops a locally solid topology from solid zero neighborhoods and proves density of , then demonstrates that the Fremlin projective tensor product preserves both -convergence and -convergence for Banach lattices, initially under mild assumptions and ultimately in full generality using a key tensor-product inequality. A central contribution is a topological framework that yields preservation results without extra hypotheses and provides a simpler proof of a main result in the literature (Z:25). Overall, the work enhances the understanding of unbounded convergence in tensor products of vector lattices and offers a robust method with potential applications in analysis on Banach lattices and locally solid vector lattices.

Abstract

Let and be locally solid vector lattices. We introduce a locally solid topology on the Fremlin tensor product and we denote it by . It extends the Fremlin projective tensor product in the setting of Banach lattices. The main purpose of the paper is to determine conditions under which, the canonical bilinear mapping is uniformly continuous when and are equipped with appropriate unbounded convergences whose associated topologies make the corresponding spaces locally solid vector lattices. By means of an inequality for tensor products in vector lattices and by exploiting the structure of the topology , we show, in particular, that the Fremlin tensor product between Banach lattices preserves several important classes of unbounded convergences.
Paper Structure (6 sections, 20 theorems, 107 equations)

This paper contains 6 sections, 20 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. motivation
  3. Preliminaries
  4. main results
  5. around preserving unbounded convergences
  6. topological approach

Key Result

Theorem 1

Suppose $E$ and $F$ are Archimedean vector lattices. Moreover, assume that $x_{\alpha}\xrightarrow{uo}x$ in $E$ and $y_{\beta}\xrightarrow{uo}y$ in $F$. Then, $x_{\alpha}\otimes y_{\beta}\xrightarrow{uo} x\otimes y$ in the Fremlin tensor product $E\overline{\otimes}F$.

Theorems & Definitions (44)

  • Theorem 1
  • Lemma 2
  • proof
  • Proposition 3
  • proof
  • Remark 4
  • Theorem 5
  • proof
  • Example 6
  • Proposition 7
  • ...and 34 more