A reduction method for noncommutative 𝐿_{𝑝}-spaces and applications

Haagerup, Uffe; Junge, Marius; Xu, Quanhua

doi:10.1090/S0002-9947-09-04935-6

A reduction method for noncommutative $L_p$-spaces and applications
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by Uffe Haagerup, Marius Junge and Quanhua Xu

Trans. Amer. Math. Soc. 362 (2010), 2125-2165

DOI: https://doi.org/10.1090/S0002-9947-09-04935-6

Published electronically: October 15, 2009

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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 The main tool is an unpublished result of the first-named author which approximates any noncommutative $L_p$-space by tracial ones.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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