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Noncommutative Boyd interpolation theorems

Author: Sjoerd Dirksen
Journal: Trans. Amer. Math. Soc. 367 (2015), 4079-4110
MSC (2010): Primary 46B70, 46L52, 46L53
Published electronically: December 10, 2014
MathSciNet review: 3324921
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Abstract: We present a new, elementary proof of Boyd's interpolation theorem. Our approach naturally yields a noncommutative version of this result and even allows for the interpolation of certain operators on $ \ell ^1$-valued noncommutative symmetric spaces. By duality we may interpolate several well-known noncommutative maximal inequalities. In particular we obtain a version of Doob's maximal inequality and the dual Doob inequality for noncommutative symmetric spaces. We apply our results to prove the Burkholder-Davis-Gundy and Burkholder-Rosenthal inequalities for noncommutative martingales in these spaces.

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Additional Information

Sjoerd Dirksen
Affiliation: Hausdorff Center for Mathematics, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Address at time of publication: Department of Mathematics, RWTH Aachen University, 52056 Aachen, Germany

Keywords: Boyd interpolation theorem, noncommutative symmetric spaces, $\Phi$-moment inequalities, Doob maximal inequality, Burkholder-Davis-Gundy inequalities, Burkholder-Rosenthal inequalities
Received by editor(s): April 12, 2012
Received by editor(s) in revised form: March 12, 2013
Published electronically: December 10, 2014
Additional Notes: This research was supported by VICI subsidy 639.033.604 of the Netherlands Organisation for Scientific Research (NWO) and the Hausdorff Center for Mathematics
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.