Duke Mathematical Journal

Operator-space Grothendieck inequalities for noncommutative $L_p$-spaces

Quanhua Xu

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We prove the operator-space Grothendieck inequality for bilinear forms on subspaces of noncommutative $L_p$-spaces with $2 \lt p \lt \infty$. One of our results states that given a map $u: E\to F^*$, where $E, F\subset L_p(M)$ ($2 \lt p \lt \infty$, $M$ 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 $L_p$-space ($2 \lt p \lt \infty$) with values in a $q$-column space for every $q\in [p', p]$ ($p'$ being the index conjugate to $p$). These results are the $L_p$-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 $L_p$-spaces ($2 \lt p \lt \infty$). In particular, we show that the norm dual to this tensor norm, when restricted to subspaces of noncommutative $L_p$-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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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

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. http://projecteuclid.org/euclid.dmj/1139232349.

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