The ``maximal" tensor product of operator spaces

Abstract

In analogy with the maximal tensor product of $C^{\ast }$-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 $E^{\ast }{\otimes }_{\mu }E=E^{\ast }{\otimes }_{\min }E$ 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.

References in corpus (1)

• A simple proof of a theorem of Kirchberg and related results on $C^{\ast }$-norms

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