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The “maximal” tensor product of operator spaces

Published online by Cambridge University Press:  20 January 2009

Timur Oikhberg  and
Gilles Pisier
Timur Oikhberg
Affiliation:
The University of Texas, Austin, Tx 78712, U.S.A.
Gilles Pisier
Affiliation:
Texas A&M UniversityCollege Station TX 77843U.S.A. and Université Paris 6, Paris, France
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In analogy with the maximal tensor product of C*-algebras, we define the “maximal” tensor product E1⊗μE2 of two operator spaces E1 and E2 and we show that it can be identified completely isometrically with the sum of the two Haagerup tensor products: E1⊗hE2 + E2⊗hE1. We also study the extension to more than two factors. Let E be an n-dimensional operator space. As an application, we show that the equality E*⊗μE = E*⊗min E holds isometrically iff E = Rn or E = Cn (the row or column n-dimensional Hilbert spaces). Moreover, we show that if an operator space E is such that, for any operator space F, we have F ⊗min E = F⊗μ E isomorphically, then E is completely isomorphic to either a row or a column Hilbert space.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 42 , Issue 2 , June 1999 , pp. 267 - 284
Copyright
Copyright © Edinburgh Mathematical Society 1999

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