On script O sign ℒ_∞ structures of nuclear C*-algebras

We study the local operator space structure of nuclear C*-algebras. It is shown that a C*-algebra is nuclear if and only if it is an script O signℒ_{∞, λ} space for some (and actually for every) λ > 6. The script O signℒ_∞ constant λ provides an interesting invariant script O signℒ_∞(A) = inf(λ: A is an script O signℒ_{∞, λ} space} for nuclear C*-algebras. Indeed, if A is a nuclear C*-algebra, then we have 1 ≤ script O signℒ_∞(A) ≤ 6, and if A is a unital nuclear C*-algebra with script O signℒ_∞(A) ≤ (1+√5/2)^1/2, we show that A must be stably finite. We also investigate the connection between the rigid script O signℒ_{∞, 1+} structure and the rigid complete order script O signℒ_{∞, 1+} structure on C*-algebras, where the latter structure has been studied by Blackadar and Kirchberg in their characterization of strong NF C*-algebras. Another main result of this paper is to show that these two local structrues are actually equivalent on unital nuclear C*-algebras. We obtain this by showing that if a unital (nuclear) C*-algebra is a rigid script O signℒ_{∞, 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*-algebra is an script O signℒ_{∞, 1+} space, then it is quasi-diagonal, and thus is an NF algebra.