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15 February 2006 Operator-space Grothendieck inequalities for noncommutative Lp-spaces
Quanhua Xu
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Duke Math. J. 131(3): 525-574 (15 February 2006). DOI: 10.1215/S0012-7094-06-13135-6

Abstract

We prove the operator-space Grothendieck inequality for bilinear forms on subspaces of noncommutative Lp-spaces with 2<p<. One of our results states that given a map u:EF*, where E,FLp(M) (2<p<, M being a von Neumann algebra), u is completely bounded if and only if u factors through a direct sum of a p-column space and a p-row space. We also obtain several operator-space versions of the classical little Grothendieck inequality for maps defined on a subspace of a noncommutative Lp-space (2<p<) with values in a q-column space for every q[p',p] (p' being the index conjugate to p). These results are the Lp-space analogues of the recent works on the operator-space Grothendieck theorems by Pisier and Shlyakhtenko. The key ingredient of our arguments is some Khintchine-type inequalities for Shlyakhtenko's generalized circular systems. One of our main tools is a Haagerup-type tensor norm that turns out to be particularly fruitful when applied to subspaces of noncommutative Lp-spaces (2<p<). In particular, we show that the norm dual to this tensor norm, when restricted to subspaces of noncommutative Lp-spaces, is equal to the factorization norm through a p-row space

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Quanhua Xu. "Operator-space Grothendieck inequalities for noncommutative Lp-spaces." Duke Math. J. 131 (3) 525 - 574, 15 February 2006. https://doi.org/10.1215/S0012-7094-06-13135-6

Information

Published: 15 February 2006
First available in Project Euclid: 6 February 2006

zbMATH: 1129.46048
MathSciNet: MR2219250
Digital Object Identifier: 10.1215/S0012-7094-06-13135-6

Subjects:
Primary: 46L07
Secondary: 46L50

Rights: Copyright © 2006 Duke University Press

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Vol.131 • No. 3 • 15 February 2006
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