The "maximal" tensor product of operator spaces

In analogy with the maximal tensor product of C*-algebras, we define the "m aximal" tensor product E-1 circle times(mu) E-2 of two operator spaces E-1 and E-2 and we show that it can be identified completely isometrically with the sum of the two Haagerup tensor products: E-1 circle times(h) E-2 + E-2 circle times(h) E-1. 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* circle times(mu) E = E* circle times(min) E holds isometric ally iff E = R-n or E = C-n (the row or column n-dimensional Hilbert spaces ). Moreover, we show that if an operator space E is such that, for any oper ator space F, we have F circle times(min) E = F circle times(mu) E isomorph ically, then E is completely isomorphic to either a row or a column Hilbert space. 1991 Mathematics subject classification: 47D15, 47D25, 46M05.