The Riesz–Kantorovich formula for lexicographically ordered spaces

Abstract

If L and M are partially ordered vector spaces, then one can consider regular linear maps from L to M, i.e. linear maps which can be written as the difference of two positive linear maps. If the space L is directed, then the space Lr(L,M) of all regular linear operators becomes a partially ordered vector space itself. We will mainly concern ourselves with the questions when the space Lr(L,M) is itself a Riesz space and how, even if it is not a Riesz space, its lattice operations work. The so-called Riesz–Kantorovich theorem gives sufficient conditions for which Lr(L,M) is a Riesz space and it also specifies the lattice operations by means of the Riesz–Kantorovich formula: if S,T∈Lr(L,M) and x∈L with x≥0 then the supremum S∨T in the point x is given by

(S∨T)(x)=sup{S(y)+T(x−y):0≤y≤x}.
It is still an open problem if whenever in a more general setting the supremum of two regular operators exists in Lr(L,M), it automatically is given by the Riesz–Kantorovich formula. Our main result concerns the special case where L is a partially ordered vector space with a strong order unit and M is a (possibly infinite) product of copies of the real line, equipped with the lexicographic ordering. It will turn out that under some mild continuity and regularity conditions the lattice operations on Lr(L,M) are indeed given by the Riesz–Kantorovich formula, even though the space Lr(L,M) is not necessarily a Riesz space.