The ``maximal" tensor product of operator spaces

Abstract

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.