Examples of non exact 1-subexponential $C^*$-algebras

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.