The “maximal” tensor product of operator spaces

Abstract

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.