On extreme extension of positive operators

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