Bilinear forms on exact operator spaces and B(H)\otimes B(H)

Let $E,F$ be exact operators (For example subspaces of the $C^*$-algebra $K(H)$ of all the compact operators on an infinite dimensional Hilbert space $H$). We study a class of bounded linear maps $u\colon E\to F^*$ which we call tracially bounded. In particular, we prove that every completely bounded (in short $c.b.$) map $u\colon E\to F^*$ factors boundedly through a Hilbert space. This is used to show that the set $OS_n$ of all $n$-dimensional operator spaces equipped with the $c.b.$ version of the Banach Mazur distance is not separable if $n>2$. As an application we show that there is more than one $C^*$-norm on $B(H)\otimes B(H)$, or equivalently that $B(H)\otimes_{\min}B(H)\not=B(H)\otimes_{\max}B(H),$ which answers a long standing open question. Finally we show that every ``maximal" operator space (in the sense of Paulsen) is not exact in the infinite dimensional case, and in the finite dimensional case, we give a lower bound for the ``exactness constant".

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