Exact operator spaces

In this paper, we study {\it operator spaces\/} in the sense of the theory developed recently by Blecher-Paulsen [BP] and Effros-Ruan [ER1]. By an operator space, we mean a closed subspace $E\subset B(H)$, with $H$ Hilbert. We will be mainly concerned here with the ``geometry'' of {\it finite dimensional\/} operator spaces. In the Banach space category, it is well known that every separable space embeds isometrically into $\ell_\infty$. Moreover, if $E$ is a finite dimensional normed space then for each $\vp>0$, there is an integer $n$ and a subspace $F\subset \ell^n_\infty$ which is $(1+\vp)$-isomorphic to $E$, i.e. there is an isomorphism $u\colon \ E\to F$ such that $\|u\|\ \|u^{-1}\|\le 1+\vp$. Here of course, $n$ depends on $\vp$, say $n=n(\vp)$ and usually (for instance if $E = \ell^k_2$) we have $n(\vp)\to \infty$ when $\vp\to 0$. Quite interestingly, it turns out that this fact is not valid in the category of operator spaces:\ although every operator space embeds completely isometrically into $B(H)$ (the non-commutative analogue of $\ell_\infty$) it is not true that a finite dimensional operator space must be close to a subspace of $M_n$ (the non-commutative analogue of $\ell^n_\infty$) for some $n$. The main object of this paper is to study this phenomenon.