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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A reduction method for noncommutative $ L_p$-spaces and applications


Authors: Uffe Haagerup, Marius Junge and Quanhua Xu
Journal: Trans. Amer. Math. Soc. 362 (2010), 2125-2165
MSC (2000): Primary 46L51, 46L07; Secondary 47L30
Published electronically: October 15, 2009
MathSciNet review: 2574890
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Abstract: We consider the reduction of problems on general noncommutative $ L_p$-spaces to the corresponding ones on those associated with finite von Neumann algebras. The main tool is an unpublished result of the first-named author which approximates any noncommutative $ L_p$-space by tracial ones. We show that under some natural conditions a map between two von Neumann algebras extends to their crossed products by a locally compact abelian group or to their noncommutative $ L_p$-spaces. We present applications of these results to the theory of noncommutative martingale inequalities by reducing most recent general noncommutative martingale/ergodic inequalities to those in the tracial case.


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Additional Information

Uffe Haagerup
Affiliation: Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark
Email: haagerup@imada.sdu.dk

Marius Junge
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email: junge@math.uiuc.edu

Quanhua Xu
Affiliation: Laboratoire de Mathématiques, Université de France-Comté, 25030 Besançon Cedex, France
Email: qxu@univ-fcomte.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04935-6
PII: S 0002-9947(09)04935-6
Keywords: Noncommutative $L_p$-spaces, finite von Neumann algebras, reduction, crossed products, extensions, martingale inequalities, ergodic inequalities
Received by editor(s): June 6, 2008
Published electronically: October 15, 2009
Additional Notes: The first author was partially supported by the Danish Natural Science Research Council
The second author was partially supported by the National Science Foundation
The third author was partially supported by the Agence Nationale de Recherche
Article copyright: © Copyright 2009 American Mathematical Society