Order isomorphisms on function spaces

Denny H. Leung¹ ; Lei Li²

¹ Department of Mathematics National University of Singapore Singapore 119076
² School of Mathematical Sciences and LPMC Nankai University Tianjin, 300071, China

Studia Mathematica, Tome 219 (2013) no. 2, pp. 123-138

Résumé

The classical theorems of Banach and Stone (1932, 1937), Gelfand and Kolmogorov (1939) and Kaplansky (1947) show that a compact Hausdorff 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 $T: A(X) \to A(Y)$ such that $f \geq 0$ if and only if $Tf \geq 0$ gives rise to a homeomorphism $h: X \to Y$ 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.

Détail

DOI : 10.4064/sm219-2-3

Keywords: classical theorems banach stone gelfand kolmogorov kaplansky compact hausdorff space uniquely determined linear isometric structure algebraic structure lattice structure respectively space paper shown rather general subspaces respectively linear bijection geq only geq gives rise homeomorphism which represented weighted composition operator three classical results mentioned above derived corollaries generalizations noncompact spaces other function spaces spaces lipschitz functions differentiable functions presented