Persistence of Banach lattices under nonlinear order isomorphisms

Persistence of Banach lattices under nonlinear order isomorphisms Ordered vector spaces E and F are said to be order isomorphic if there is a (not necessarily linear) bijection $$T: E\rightarrow F$$ T : E → F such that $$x\ge y$$ x ≥ y if and only if $$Tx \ge Ty$$ T x ≥ T y for all $$x,y \in E$$ x , y ∈ E . We investigate some situations under which an order isomorphism between two Banach lattices implies the persistence of some linear lattice structure. For instance, it is shown that if a Banach lattice E is order isomorphic to C(K) for some compact Hausdorff space K, then E is (linearly) isomorphic to C(K) as a Banach lattice. Similar results hold for Banach lattices order isomorphic to $$c_{0}$$ c 0 , and for Banach lattices that contain a closed sublattice order isomorphic to $$c_{0}$$ c 0 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Persistence of Banach lattices under nonlinear order isomorphisms

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Publisher
Springer International Publishing
Copyright
Copyright © 2015 by Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-015-0382-0
Publisher site
See Article on Publisher Site

Abstract

Ordered vector spaces E and F are said to be order isomorphic if there is a (not necessarily linear) bijection $$T: E\rightarrow F$$ T : E → F such that $$x\ge y$$ x ≥ y if and only if $$Tx \ge Ty$$ T x ≥ T y for all $$x,y \in E$$ x , y ∈ E . We investigate some situations under which an order isomorphism between two Banach lattices implies the persistence of some linear lattice structure. For instance, it is shown that if a Banach lattice E is order isomorphic to C(K) for some compact Hausdorff space K, then E is (linearly) isomorphic to C(K) as a Banach lattice. Similar results hold for Banach lattices order isomorphic to $$c_{0}$$ c 0 , and for Banach lattices that contain a closed sublattice order isomorphic to $$c_{0}$$ c 0 .

Journal

PositivitySpringer Journals

Published: Nov 9, 2015

References

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