On 𝒪ℒ_∞ structures of nuclear C ^*-algebras

Abstract

 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 𝒪ℒ_∞,λ space for some (and actually for every) λ>6. The 𝒪ℒ_∞ constant λ provides an interesting invariant

for nuclear C ^*-algebras. Indeed, if 𝒜 is a nuclear C ^*-algebra, then we have 1≤𝒪ℒ_∞(𝒜)≤6, and if 𝒜 is a unital nuclear C ^*-algebra with , we show that 𝒜 must be stably finite. We also investigate the connection between the rigid 𝒪ℒ_∞,1+ structure and the rigid complete order 𝒪ℒ_∞,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 𝒪ℒ_∞,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 𝒪ℒ_∞,1+ space, then it is quasi-diagonal, and thus is an NF algebra.