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The “maximal” tensor product of operator spaces

Published online by Cambridge University Press:  20 January 2009

Timur Oikhberg
The University of Texas, Austin, Tx 78712, U.S.A.
Gilles Pisier
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 E1μE2 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: E1hE2 + E2hE1. 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.

Research Article
Copyright © Edinburgh Mathematical Society 1999



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