$u\tau$-continuous operators on locally solid vector lattices

TL;DR

This paper studies the structure of continuous operators under the unbounded topology on locally solid vector lattices, exploring conditions for the space to form a band and properties of operator moduli.

Contribution

It introduces the concept of $u\tau$-continuity for operators and analyzes when the space of such operators forms a band, including conditions on operator moduli.

Findings

The space of $u\tau$-continuous operators can form a band under certain conditions.

Conditions are identified under which the modulus of a continuous operator shares the $u\tau$-continuity property.

Examples illustrate the theoretical results and clarify the context.

Abstract

In this note, we consider the space of all continuous operators with respect to the unbounded topology on locally solid vector lattices. We investigate whether this space forms a band. In addition, we look into some situations under which, the modulus of a continuous operator has the same property. Some examples are given to make the context more understandable.