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Weak Banach-Saks property and Komlós' theorem for preduals of JBW$ ^*$-triples

Authors: Antonio M. Peralta and Hermann Pfitzner
Journal: Proc. Amer. Math. Soc. 144 (2016), 4723-4731
MSC (2010): Primary 46L05, 46L40
Published electronically: July 7, 2016
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Abstract: We show that the predual of a JBW$ ^*$-triple has the weak Banach-Saks property, that is, reflexive subspaces of a JBW$ ^*$-triple predual are super-reflexive. We also prove that JBW$ ^*$-triple preduals satisfy the Komlós property (which can be considered an abstract version of the weak law of large numbers). The results rely on two previous papers from which we infer the fact that, like in the classical case of $ L^1$, a subspace of a JBW$ ^*$-triple predual contains $ \ell _1$ as soon as it contains uniform copies of $ \ell _1^n$.

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Antonio M. Peralta
Affiliation: Departamento de Análisis Matemático, Universidad de Granada, Facultad de Ciencias 18071, Granada, Spain

Hermann Pfitzner
Affiliation: Laboratoire de mathématiques MAPMO UMR 7349, Université d’Orléans, BP 6759, F-45067 Orléans Cedex 2, France

Received by editor(s): May 20, 2015
Published electronically: July 7, 2016
Additional Notes: The first author was partially supported by the Spanish Ministry of Economy and Competitiveness and European Regional Development Fund project no. MTM2014-58984-P and Junta de Andalucía grant FQM375.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2016 American Mathematical Society