Duke Mathematical Journal

Operator-space Grothendieck inequalities for noncommutative Lp-spaces

Quanhua Xu

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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

Article information

Duke Math. J., Volume 131, Number 3 (2006), 525-574.

First available in Project Euclid: 6 February 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L07: Operator spaces and completely bounded maps [See also 47L25]
Secondary: 46L50


Xu, Quanhua. Operator-space Grothendieck inequalities for noncommutative $L_p$ -spaces. Duke Math. J. 131 (2006), no. 3, 525--574. doi:10.1215/S0012-7094-06-13135-6. https://projecteuclid.org/euclid.dmj/1139232349

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