  This is a complement to our previous paper on the arxiv on quantum expanders
and geometry of operator spaces. We show that there is a non-exact
$C^*$-algebra that is 1-subexponential, and we give several other complements
to the results of that paper. Our example can be described very simply using
random matrices: Let ${X_j^{(m)}\mid j=1,2,...}$ be an i.i.d. sequence of
random $m\times m$-matrices distributed according to the Gaussian Unitary
Ensemble (GUE). For each $j$ let $u_j(\omega)$ be the block direct sum defined
by $$u_j(\omega)= \oplus_{m\ge 1} X_j^{(m)}(\omega)\in \oplus_{m\ge 1} M_m.$$
Then for almost every $\omega$ the $C^*$-algebra generated by ${u_j(\omega)
\mid j=1,2,...}$ is 1-subexponential but is not exact. The GUE is a matrix
model for the semi-circular distribution. We can also use instead the analogous
circular model.
