  In analogy with the maximal tensor product of $C^*$-algebras, we define the
``maximal" tensor product $E_1\otimes_\mu E_2$ 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: \ $E_1\otimes_h E_2 + E_2\otimes_h
E_1$. Let $E$ be an $n$-dimensional operator space. As an application, we show
that the equality $E^* \otimes_\mu E=E^* \otimes_{\rm min} E$ holds
isometrically 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 operator space $F$, we have $F\otimes_{\min} E=F\otimes_{\mu} E$
isomorphically, then $E$ is completely isomorphic to either a row or a column
Hilbert space.
