On ${\text{OL}}_{\mathrm{\infty }}$ structure of nuclear $C^{\ast }$-algebras

We study the local operator space structure of nuclear $C^{\ast }$-algebras. It is shown that a $C^{\ast }$-algebra is nuclear if and only if it is an $\OL_{\infty, \la}$spaceforsome(andactuallyforevery)$\la > 6$\mathrm{.}The$\OL_\infty$constant$\lambda$ provides an interesting invariant $${\text{OL}}_{\mathrm{\infty }}(\text{A})=\inf \{\text{la}:\text{A}\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{n}{\text{OL}}_{\mathrm{\infty },\text{la}}\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}\}$$ for nuclear $C^{\ast }$-algebras. Indeed, if $\text{A}$ is a nuclear $C^{\ast }$-algebra, then we have $1\le {\text{OL}}_{\mathrm{\infty }}(\text{A})\le 6$, and if $\text{A}$ is a unital nuclear $C^{\ast }$-algebra with ${\text{OL}}_{\mathrm{\infty }}(\text{A})\le (\frac{1+\sqrt{5}}{2}{)}^{\frac{1}{2}}$, we show that $\text{A}$ must be stably finite. We also investigate the connection between the rigid ${\text{OL}}_{\mathrm{\infty },1^{+}}$ structure and the rigid complete order ${\text{OL}}_{\mathrm{\infty },1^{+}}$ structure on $C^{\ast }$-algebras, where the latter structure has been studied by Blackadar and Kirchberg in their characterization of strong NF $C^{\ast }$-algebras. Another main result of this paper is to show that these two local structrues are actually equivalent on unital nuclear $C^{\ast }$-algebras. We obtain this by showing that if a unital (nuclear) $C^{\ast }$-algebra is a rigid ${\text{OL}}_{\mathrm{\infty },1^{+}}$ space, then it is inner quasi-diagonal, and thus is a strong NF algebra. It is also shown that if a unital (nuclear) $C^{\ast }$-algebra is an ${\text{OL}}_{\mathrm{\infty },1^{+}}$ space, then it is quasi-diagonal, and thus is an NF algebra.