Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

The “maximal” tensor product of operator spaces

Timur Oikhberga1 and Gilles Pisiera2*

a1 The University of Texas, Austin, Tx 78712, U.S.A.

a2 Texas A&M University College Station TX 77843 U.S.A. and Université Paris 6, Paris, France

In analogy with the maximal tensor product of C*-algebras, we define the “maximal” tensor product E1xs2297μ E2 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: E1xs2297hE2 + E2xs2297hE1. 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*xs2297μE = E*xs2297min 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 xs2297min E = Fxs2297μ E isomorphically, then E is completely isomorphic to either a row or a column Hilbert space.

(Received February 12 1997)


* Partially supported by the N.S.F.