Order isomorphisms on function spaces

Abstract

The classical theorems of Banach and Stone (1932, 1937), Gelfand and Kolmogorov (1939) and Kaplansky (1947) show that a compact Hausdor space X is uniquely determined by the linear isometric structure, the algebraic structure, and the lattice structure, respectively, of the space C(X). In this paper, it is shown that for rather general subspaces A(X) and A(Y ) of C(X) and C(Y ), respectively, any linear bijection T : A(X) → A(Y ) such that f ≤ 0 if and only if Tf ≤ 0 gives rise to a homeomorphism h : X ! Y with which T can be represented as a weighted composition operator. The three classical results mentioned above can be derived as corollaries. Generalizations to noncompact spaces and other function spaces such as spaces of Lipschitz functions and differentiable functions are presented. © 2013 Instytut Matematyczny PAN.