The Grothendieck property from an ordered point of view

TL;DR

This paper introduces the unbounded Grothendieck property in Banach lattices and proves it characterizes reflexive Banach lattices, offering a new ordered perspective on the classical property.

Contribution

It defines the unbounded Grothendieck property in Banach lattices and establishes its equivalence with reflexivity, providing a novel ordered framework for the property.

Findings

Spaces with the unbounded Grothendieck property are exactly reflexive Banach lattices.

The unbounded Grothendieck property generalizes the classical Grothendieck property in an ordered setting.

Abstract

In this note, we consider several notions related to the Grothendieck property. Among them, we introduce the notion "unbounded Grothendieck property" in a Banach lattice as an unbounded version of the known Grothedieck property in the Banach space theory. Beside other results, surprisingly, we show that spaces with the unbounded Grothendieck property are exactly the reflexive Banach lattices.