On the Linearity of Order-isomorphisms

Canadian Journal of Mathematics, vol. 73, pp. 399–416

Abstract

Abstract A basic problem in the theory of partially ordered vector spaces is to characterise those cones on which every order-isomorphism is linear. We show that this is the case for every Archimedean cone that equals the inf-sup hull of the sum of its engaged extreme rays. This condition is milder than existing ones and is satisfied by, for example, the cone of positive operators in the space of bounded self-adjoint operators on a Hilbert space. We also give a general form of order-isomorphisms on the inf-sup hull of the sum of all extreme rays of the cone, which extends results of Artstein–Avidan and Slomka to infinite-dimensional partially ordered vector spaces, and prove the linearity of homogeneous order-isomorphisms in a variety of new settings.

Authors 2

1. Bas Lemmens

   University of Kent

   Affiliation as printed

   School of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury, CT2 7NX, United Kingdom Email:

   University of Kent:

2. Onno van Gaans Aachen Mathematical Institute

   Leiden University

   Affiliation as printed

   Mathematical Institute, Leiden University, P.O.Box 9512, 2300 RALeiden, The Netherlands Email:

   Leiden University