Mathematische Annalen

, Volume 325, Issue 3, pp 449–483

On 𝒪ℒ structures of nuclear C*-algebras

  • Marius Junge
  • Narutaka Ozawa
  • Zhong-Jin Ruan

DOI: 10.1007/s00208-002-0384-7

Cite this article as:
Junge, M., Ozawa, N. & Ruan, ZJ. Math. Ann. (2003) 325: 449. doi:10.1007/s00208-002-0384-7


 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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Marius Junge
    • 1
  • Narutaka Ozawa
    • 2
  • Zhong-Jin Ruan
    • 3
  1. 1.Department of Mathematics, University of Illinois, Urbana, IL 61801, USA (e-mail:
  2. 2.Department of Mathematical Sciences, University of Tokyo, Komaba, 153-8914, Japan (e-mail:
  3. 3.Department of Mathematics, University of Illinois, Urbana, IL 61801, USA (e-mail: