On ∞ structure of nuclear C*-algebras

Original · EN

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 ∞, space for some (and actually for every) > 6. The ∞ constant λ provides an interesting invariant ∞ () = {: is an ∞, space} for nuclear C*-algebras. Indeed, if is a nuclear C*-algebra, then we have 1≤ ∞ () ≤ 6, and if is a unital nuclear C*-algebra with ∞ () ≤ (1+√ 52) 12, we show that must be stably finite. We also investigate the connection between the rigid ∞, ₁₊ structure and the rigid complete order ∞, 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 ∞, 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 ∞, ₁₊ space, then it is quasi-diagonal, and thus is an NF algebra.