Duke Mathematical Journal
- Duke Math. J.
- Volume 131, Number 3 (2006), 525-574.
Operator-space Grothendieck inequalities for noncommutative -spaces
We prove the operator-space Grothendieck inequality for bilinear forms on subspaces of noncommutative -spaces with . One of our results states that given a map , where (, being a von Neumann algebra), is completely bounded if and only if factors through a direct sum of a -column space and a -row space. We also obtain several operator-space versions of the classical little Grothendieck inequality for maps defined on a subspace of a noncommutative -space () with values in a -column space for every ( being the index conjugate to ). These results are the -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 -spaces (). In particular, we show that the norm dual to this tensor norm, when restricted to subspaces of noncommutative -spaces, is equal to the factorization norm through a -row space
Duke Math. J., Volume 131, Number 3 (2006), 525-574.
First available in Project Euclid: 6 February 2006
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 46L07: Operator spaces and completely bounded maps [See also 47L25]
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