On ${\mathcal{O}\mathcal{L}}_{\mathrm{\infty }}$ structure of nuclear, quasidiagonal C*-algebras

We continue the study of ${\mathcal{O}\mathcal{L}}_{\mathrm{\infty }}$ structure of nuclear $C^{\ast }$-algebras initiated by Junge, Ozawa and Ruan. In particular, we prove if ${\mathcal{O}\mathcal{L}}_{\mathrm{\infty }}(A)<1.005,$ then $A$ has a separating family of irreducible, stably finite representations. As an application we give examples of nuclear, quasidiagonal $C^{\ast }$-algebras $A$ with ${\mathcal{O}\mathcal{L}}_{\mathrm{\infty }}(A)>1.$