Fremlin tensor product behaves well with the unbounded order convergence

TL;DR

The paper investigates whether the Fremlin tensor product $E\\overline{\\otimes}F$ preserves order convergence ($o$) and unbounded order convergence ($uo$) for Archimedean vector lattices. It uses the Buskes–Wickstead $S(\\Sigma)$-space framework and Maeda–Ogasawara theory to realize vector lattices as sublattices and to analyze convergence in $S(\\Sigma\\times\\Omega)$. The main contributions provide a joint extension of Grobler's componentwise results, proving that nets that converge $uo$ in each factor yield $uo$-convergence of their tensor product, and that $o$-convergence is preserved under tensoring when nets are order null and one is eventually order bounded; the latter assumption is shown to be essential. These results clarify how convergence notions transfer through Fremlin tensor products, with implications for analysis on vector lattices and operator theory.

Abstract

Suppose $Σ$ is a topological space and $S(Σ)$ is the vector lattice of all equivalent classes of continuous real-valued functions defined on open dense subsets of $Σ$. In this paper, we establish some lattice and topological aspects of $S(Σ)$. In particular, as an application, we show that the unbounded order convergence and the order convergence are stable under passing to the Fremlin tensor product of two Archimedean vector lattices.

Theorems & Definitions (15)

• Lemma 1
• Corollary 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• Theorem 7