Abstract
Unbounded order convergence has lately been systematically studied as a generalization of almost everywhere convergence to the abstract setting of vector and Banach lattices. This paper presents a duality theory for unbounded order convergence. We define the unbounded order dual (or uo-dual) \({X_{uo}^\sim }\) of a Banach lattice X and identify it as the order continuous part of the order continuous dual \({X_n^\sim }\). The result allows us to characterize the Banach lattices that have order continuous preduals and to show that an order continuous predual is unique when it exists. Applications to the Fenchel–Moreau duality theory of convex functionals are given. The applications are of interest in the theory of risk measures in Mathematical Finance.
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Niushan Gao was a PIMS Postdoctoral Fellow at the University of Lethbridge when this work was done.
Denny H. Leung is partially supported by AcRF grant R-146-000-242-114.
Foivos Xanthos acknowledges the support of an NSERC grant.
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Gao, N., Leung, D.H. & Xanthos, F. Duality for unbounded order convergence and applications. Positivity 22, 711–725 (2018). https://doi.org/10.1007/s11117-017-0539-0
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DOI: https://doi.org/10.1007/s11117-017-0539-0
Keywords
- Unbounded order dual
- Order continuous dual
- Order continuous predual
- Monotonically complete Banach lattices
- Representation of convex functionals