On Extreme Extension of Positive Operators

Abstract

Given vector lattices $E$, $F$ and a positive operator $S$ from a majorzing subspace $D$ of $E$ to~$F$, denote by $\mathcal{E}(S)$ the collection of all positive extension of $S$ to all of $E$. This note aims to describe the collection of extreme points of the convex set $\mathcal{E}(T\circ S)$. It is proved, in particular, that $\mathcal{E}(T\circ S)$ and $T\circ\mathcal{E}(S)$ coincide and every extreme point of $\mathcal{E}(T\circ S)$ is an~extreme point of $T\circ\mathcal{E}(S)$, whenever $T:F\to G$ is a~Maharam operator between Dedekind complete vector lattices. The proofs of the main results are based on three ingredients: a characterization of extreme points of subdifferentials, abstract disintegration in~Kantorovich spaces, and an intrinsic characterization of subdifferentials.