Order isomorphisms on function spaces

Denny H. Leung, and Lei Li. "Order isomorphisms on function spaces." Studia Mathematica 219.2 (2013): 123-138. <http://eudml.org/doc/285469>.

@article{DennyH2013,
abstract = {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) → 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.},
author = {Denny H. Leung, Lei Li},
journal = {Studia Mathematica},
keywords = {linear-order isomorphisms; subspaces of spaces of continuous functions; Banach-Stone theorem},
language = {eng},
number = {2},
pages = {123-138},
title = {Order isomorphisms on function spaces},
url = {http://eudml.org/doc/285469},
volume = {219},
year = {2013},
}

TY - JOUR
AU - Denny H. Leung
AU - Lei Li
TI - Order isomorphisms on function spaces
JO - Studia Mathematica
PY - 2013
VL - 219
IS - 2
SP - 123
EP - 138
AB - 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) → 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.
LA - eng
KW - linear-order isomorphisms; subspaces of spaces of continuous functions; Banach-Stone theorem
UR - http://eudml.org/doc/285469
ER -