On 𝒪ℒ∞ structures of nuclear C*-algebras
- Cite this article as:
- Junge, M., Ozawa, N. & Ruan, ZJ. Math. Ann. (2003) 325: 449. doi:10.1007/s00208-002-0384-7
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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.
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