The Grothendieck property from an ordered point of view

Abstract

In this note, we consider several notions related to the Grothendieck property. Among them, we introduce the notion of the unbounded Grothendieck property for Banach lattices 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.