  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".
