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On the Linearity of Order-isomorphisms

Part of: Normed linear spaces and Banach spaces; Banach lattices Special matrices Nonlinear operators and their properties

Published online by Cambridge University Press:  03 January 2020

Bas Lemmens ,
Onno van Gaans  and
Hendrik van Imhoff
Bas Lemmens
Affiliation:
School of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury, CT2 7NX, United Kingdom Email: B.Lemmens@kent.ac.uk
Onno van Gaans
Affiliation:
Mathematical Institute, Leiden University, P.O.Box 9512, 2300 RALeiden, The Netherlands Email: vangaans@math.leidenuniv.nlhvanimhoff@gmail.com
Hendrik van Imhoff
Affiliation:
Mathematical Institute, Leiden University, P.O.Box 9512, 2300 RALeiden, The Netherlands Email: vangaans@math.leidenuniv.nlhvanimhoff@gmail.com
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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.

Keywords

order-isomorphismaffine mapinf-sup hull

MSC classification

Primary: 46B40: Ordered normed spaces
Secondary: 15B48: Positive matrices and their generalizations; cones of matrices 47H07: Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 2 , April 2021 , pp. 399 - 416
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Copyright
© Canadian Mathematical Society 2020

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