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Decomposable approximations and approximately finite dimensional C*-algebras

Published online by Cambridge University Press:  26 May 2016

JORGE CASTILLEJOS*
Affiliation:
School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QW. e-mail: [email protected]

Abstract

Nuclear C*-algebras having a system of completely positive approximations formed with convex combinations of a uniformly bounded number of order zero summands are shown to be approximately finite dimensional.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1] Blackadar, B. Operator Algebras Encyclopaedia of Mathematical Sciences vol. 122 (Springer-Verlag, Berlin, 2006). Theory of C*-algebras and von Neumann algebras, Operator Algebras and Non-commutative Geometry, III.CrossRefGoogle Scholar
[2] Bratteli, O. Inductive limits of finite dimensional C*-algebras. Trans. Amer. Math. Soc. 171 (1972), 195234.Google Scholar
[3] Busby, R. Double centralizers and extensions of C*-algebras. Trans. Amer. Math. Soc. 132 (1968), 7999.Google Scholar
[4] Choi, M. D. and Effros, E. Nuclear C*-algebras and the approximation property. Amer. J. Math. 100 (1) (1978), 6179.Google Scholar
[5] Elliott, G. and Toms, A. Regularity properties in the classification program for separable amenable C*-algebras. Bull. Amer. Math. Soc. 45 (2) (2008), 229245.CrossRefGoogle Scholar
[6] Farah, I. and Katsura, T. Nonseparable UHF algebras I: Dixmier's problem. Adv. Math. 225 (3) (2010), 13991430.Google Scholar
[7] Hirshberg, I., Kirchberg, E. and White, S. Decomposable approximations of nuclear C*-algebras. Adv. Math. 230 (3) (2012), 10291039.Google Scholar
[8] Kirchberg, E. C*-nuclearity implies CPAP. Math. Nachr. 76 (1977), 203212.CrossRefGoogle Scholar
[9] Kirchberg, E. and Winter, W. Covering dimension and quasidiagonality. Internat. J. Math. 15 (1) (2004), 6385.CrossRefGoogle Scholar
[10] Matui, H. and Sato, Y. Strict comparison and $\mathcal{Z}$ -absorption of nuclear C*-algebras. Acta Math. 209 (1) (2012), 179196.Google Scholar
[11] Matui, H. and Sato, Y. Decomposition rank of UHF-absorbing C*-algebras. Duke Math. J. 163 (14) (2014), 26872708.Google Scholar
[12] Robert, L. Nuclear dimension and n-comparison. Münster J. Math. 4 (2011), 6571.Google Scholar
[13] Sato, Y., White, S. and Winter, W. Nuclear dimension and $\mathcal{Z}$ -stability. Invent. Math. 202 (2) (2015), 893921.Google Scholar
[14] Tikuisis, A. and Winter, W. Decomposition rank of $\mathcal{Z}$ -stable C*-algebras. Anal. PDE. 7 (3) (2014), 673700.CrossRefGoogle Scholar
[15] Winter, W. Covering dimension for nuclear C*-algebras. J. Funct. Anal. 199 (2) (2003), 535556.CrossRefGoogle Scholar
[16] Winter, W. Decomposition rank and $\mathcal{Z}$ -stability. Invent. Math. 179 (2) (2010), 229301.Google Scholar
[17] Winter, W. Nuclear dimension and $\mathcal{Z}$ -stability of pure C*-algebras. Invent. Math. 187 (2) (2012), 259342.Google Scholar
[18] Winter, W. and Zacharias, J. Completely positive maps of order zero. Münster J. Math. 2 (2009), 311324.Google Scholar
[19] Winter, W. and Zacharias, J. The nuclear dimension of C*-algebras. Adv. Math. 224 (2) (2010), 461498.Google Scholar
[20] Wolff, M. Disjointness preserving operators on C*-algebras. Arch. Math. (Basel) 62 (3) (1994), 248253.Google Scholar