The "maximal" tensor product of operator spaces

In analogy with the maximal tensor product of C-algebras, we define the "maximal" tensor product £, &f £2 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 ⊗ E₂ + E₂ ⊗ E₁. We also study the extension to more than two factors. Let £be an n-dimensional operator space. As an application, we show that the equality E ⊗ E = E ⊗_min E holds isometrically iff £= R, or E = C, (the row or column n-dimensional Hubert spaces). Moreover, we show that if an operator space £is such that, for any operator space F, we have F ⊗ £= F ⊗, £isomorphically, then £is completely isomorphic to either a row or a column Hubert space.