Order isomophisms between Riesz spaces

B L Van Engelen,A C M Van Rooij

doi:10.1007/s11117-018-0560-y

Abstract

The first aim of this paper is to give a description of the (not necessarily linear) order isomorphisms C(X)rightarrow C(Y) where X, Y are compact Hausdorff spaces. For a simple case, suppose X is metrizable and T is such an order isomorphism. By a theorem of Kaplansky, T induces a homeomorphism tau :Xrightarrow Y. We prove the existence of a homeomorphism Xtimes mathbb {R}rightarrow Ytimes mathbb {R} that maps the graph of any fin C(X) onto the graph of Tf. For nonmetrizable spaces the result is similar, although slightly more complicated. Secondly, we let X and Y be compact and extremally disconnected. The theory of the first part extends directly to order isomorphisms C^{infty }(X)rightarrow C^{infty }(Y). (Here C^{infty }(X) is the space of all continuous functions Xrightarrow [-infty ,infty ] that are finite on a dense set.) The third part of the paper considers order isomorphisms T between arbitrary Archimedean Riesz spaces E and F. We prove that such a T extends uniquely to an order isomorphism between their universal completions. (In the absence of linearity this is not obvious.) It follows, that there exist an extremally disconnected compact Hausdorff space X, Riesz isomorphisms hat{} of E and F onto order dense Riesz subspaces of C^{infty }(X) and an order isomorphism S:C^{infty }(X)rightarrow C^{infty }(X) such that hat{Tf}=Shat{f} (fin E).

Highlights

Two Riesz spaces may be order isomorphic without being Riesz isomorphic, an example being formed by l1 and l2
Order isomorphic Riesz spaces necessarily share certain properties that at first sight seem to depend on the Riesz space structure and on the ordering
We start with the Riesz spaces C(X ), C(Y ) where X and Y are compact Hausdorff spaces

Summary

Introduction

Two Riesz spaces may be order isomorphic without being Riesz isomorphic, an example being formed by l1 and l2. If two Archimedean Riesz spaces are order isomorphic, their universal completions (in the sense of [1]) are order and even. The definition of the universal completion involves the vector space structure. In this paper we consider order isomorphisms between Riesz spaces. We show that such a T extends uniquely to an order isomorphism between the universal completions of E and F. These universal completions are Riesz isomorphic to some C∞(X ) and we can apply the results of Sect. For s ∈ R we denote by s the constant function with value s (on any given set)

Kaplansky’s theorem

Vertically acting order isomorphisms

Arbitrary Archimedean Riesz spaces