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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Convergent martingales of operators and the Radon Nikodým property in Banach spaces


Authors: Stuart F. Cullender and Coenraad C. A. Labuschagne
Journal: Proc. Amer. Math. Soc. 136 (2008), 3883-3893
MSC (2000): Primary 46B28, 47B60, 60G48
Published electronically: June 24, 2008
MathSciNet review: 2425728
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Abstract: We extend Troitsky's ideas on measure-free martingales on Banach lattices to martingales of operators acting between a Banach lattice and a Banach space. We prove that each norm bounded martingale of cone absolutely summing (c.a.s.) operators (also known as $ 1$-concave operators), from a Banach lattice $ E$ to a Banach space $ Y$, can be generated by a single c.a.s.  operator. As a consequence, we obtain a characterization of Banach spaces with the Radon Nikodým property in terms of convergence of norm bounded martingales defined on the Chaney-Schaefer $ l$-tensor product $ E\widetilde{\otimes}_l Y$. This extends a classical martingale characterization of the Radon Nikodým property, formulated in the Lebesgue-Bochner spaces $ L^p(\mu,Y)$ ( $ 1< p <\infty$).


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Additional Information

Stuart F. Cullender
Affiliation: School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, P.O. WITS 2050, South Africa
Email: scullender@gmail.com

Coenraad C. A. Labuschagne
Affiliation: School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, P.O. WITS 2050, South Africa
Email: Coenraad.Labuschagne@wits.ac.za

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09537-3
PII: S 0002-9939(08)09537-3
Keywords: Bochner norm, Radon Nikod\'ym property, convergent martingale, cone absolutely summing operator, $1$-concave operator, Banach space, Banach lattice.
Received by editor(s): August 1, 2007
Published electronically: June 24, 2008
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2008 American Mathematical Society