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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Convergent martingales of operators and the Radon Nikodým property in Banach spaces
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by Stuart F. Cullender and Coenraad C. A. Labuschagne PDF
Proc. Amer. Math. Soc. 136 (2008), 3883-3893 Request permission

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
  • Received by editor(s): August 1, 2007
  • Published electronically: June 24, 2008
  • Communicated by: N. Tomczak-Jaegermann
  • © Copyright 2008 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 136 (2008), 3883-3893
  • MSC (2000): Primary 46B28, 47B60, 60G48
  • DOI: https://doi.org/10.1090/S0002-9939-08-09537-3
  • MathSciNet review: 2425728