Random matrices and subexponential operator spaces

We introduce and study a generalization of the notion of exact operator space that we call subexponential. Using Random Matrices we show that the factorization results of Grothendieck type that are known in the exact case all extend to the subexponential case, but we exhibit (a continuum of distinct) examples of non-exact subexponential operator spaces, as well as a C*-algebra that is subexponential with constant 1 but not exact. We also show that OH, R + C and max(?₂) (or any other maximal operator space) are not subexponential.