On script O sign ℒ structures of nuclear C*-algebras

Marins Junge, Narutaka Ozawa, Zhong Jin Ruan

Research output: Contribution to journalArticlepeer-review

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 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.

Original languageEnglish (US)
Pages (from-to)449-483
Number of pages35
JournalMathematische Annalen
Volume325
Issue number3
DOIs
StatePublished - Mar 1 2003

ASJC Scopus subject areas

  • Mathematics(all)

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