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

TL;DR

This paper introduces a new locally solid topology on the Fremlin tensor product of locally solid vector lattices, extending the concept in Banach lattices and ensuring Hausdorff properties under certain conditions.

Contribution

It establishes a Hausdorff locally solid topology on the Fremlin tensor product for locally solid vector lattices, generalizing existing tensor product topologies.

Findings

The topology extends the Fremlin projective tensor product in Banach lattices.

The topology is Hausdorff if both lattices are Hausdorff.

Provides a framework for tensor products in locally solid vector lattices.

Abstract

Let $E$ and $F$ be locally solid vector lattices. In this short note, we establish a locally solid topology on the Fremlin tensor product $E\overline{\otimes}F$ and we denote it by $\tau_{E\overline{\otimes}F}$. It extends the Fremlin projective tensor product in the setting of Banach lattices. We show that $\tau_{E\overline{\otimes}F}$ is Hausdorff provided that both $E$ and $F$ are Hausdorff.