[Banach] Abstract of a paper by Denny H. Leung and Lei Li

This is an announcement for the paper "Order isomorphisms on function spaces" by Denny H. Leung and Lei Li.

Abstract: The classical theorems of Banach and Stone, Gelfand and Kolmogorov, and Kaplansky 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 uniformly continuous functions, Lipschitz functions and differentiable functions are presented.

Archive classification: math.FA

Mathematics Subject Classification: 46E15

Submitted from: matlhh@nus.edu.sg

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1310.7351

or

http://arXiv.org/abs/1310.7351