Order isomophisms between Riesz spaces

摘要

The first aim of this paper is to give a description of the (not necessarily linear) order isomorphisms C(X)→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 τ:X→Y. We prove the existence of a homeomorphism X×R→Y×R that maps the graph of any f∈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∞(X)→C∞(Y). (Here C∞(X) is the space of all continuous functions X→[-∞,∞] 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 ^ of E and F onto order dense Riesz subspaces of C∞(X) and an order isomorphism S:C∞(X)→C∞(X) such that Tf^=Sf^ (f∈E).