Abstract
I present a new proof of Kirchberg’s
Funding statement: This work was partially funded by the Carlsberg Foundation through an Internationalisation Fellowship.
Acknowledgements
Parts of this paper were completed during a research visit at the Mittag–Leffler Institute during the programme Classification of Operator Algebras: Complexity, Rigidity, and Dynamics, and during a visit at the CRM institute during the programme IRP Operator Algebras: Dynamics and Interactions. I am thankful for their hospitality during these visits. I am very grateful to Joan Bosa, Jorge Castillejos, Aidan Sims and Stuart White for valuable, inspiring conversations on topics of the paper. I would also like to thank the referee for many helpful comments and suggestions.
References
[1]
P. Ara, F. Perera and A. S. Toms,
K-theory for operator algebras. Classification of
[2]
E. Blanchard,
Déformations de
[3]
E. Blanchard and E. Kirchberg,
Non-simple purely infinite
[4]
J. Bosa, N. P. Brown, Y. Sato, A. Tikuisis, S. White and W. Winter,
Covering dimension of
[5]
O. Bratteli and G. A. Elliott,
Structure spaces of approximately finite-dimensional
[6]
J. Brown, L. O. Clark and A. Sierakowski,
Purely infinite
[7]
N. P. Brown and N. Ozawa,
[8]
A. Connes,
Une classification des facteurs de type
[9]
K. T. Coward, G. A. Elliott and C. Ivanescu,
The Cuntz semigroup as an invariant for
[10]
J. Cuntz,
K-theory for certain
[11] M. Dadarlat, On the topology of the Kasparov groups and its applications, J. Funct. Anal. 228 (2005), no. 2, 394–418. 10.1016/j.jfa.2005.02.015Search in Google Scholar
[12] M. Dadarlat, The homotopy groups of the automorphism group of Kirchberg algebras, J. Noncommut. Geom. 1 (2007), no. 1, 113–139. 10.4171/JNCG/3Search in Google Scholar
[13]
M. Dadarlat,
Fiberwise KK-equivalence of continuous fields of
[14] M. Dadarlat and T. A. Loring, A universal multicoefficient theorem for the Kasparov groups, Duke Math. J. 84 (1996), no. 2, 355–377. 10.1215/S0012-7094-96-08412-4Search in Google Scholar
[15]
M. Dadarlat and W. Winter,
On the KK-theory of strongly self-absorbing
[16]
G. A. Elliott,
On the classification of
[17]
G. A. Elliott, G. Gong, H. Lin and Z. Niu,
On the classification of simple
[18] G. A. Elliott and D. Kucerovsky, An abstract Voiculescu–Brown–Douglas–Fillmore absorption theorem, Pacific J. Math. 198 (2001), no. 2, 385–409. 10.2140/pjm.2001.198.385Search in Google Scholar
[19] J. Gabe, Lifting theorems for completely positive maps, preprint (2016), https://arxiv.org/abs/1508.00389v3. 10.4171/JNCG/479Search in Google Scholar
[20]
G. Gong, H. Lin and Z. Niu,
Classification of finite simple amenable
[21] H. Harnisch and E. Kirchberg, The inverse problem for primitive ideal spaces, Preprintreihe SFB 478 (2005). Search in Google Scholar
[22]
G. G. Kasparov,
Hilbert
[23]
E. Kirchberg,
Exact
[24]
E. Kirchberg,
On restricted perturbations in inverse images and a description of normalizer algebras in
[25]
E. Kirchberg,
Das nicht-kommutative Michael-Auswahlprinzip und die Klassifikation nicht-einfacher Algebren,
[26]
E. Kirchberg,
Central sequences in
[27]
E. Kirchberg,
The classification of purely infinite
[28]
E. Kirchberg and N. C. Phillips,
Embedding of exact
[29]
E. Kirchberg and M. Rørdam,
Non-simple purely infinite
[30]
E. Kirchberg and M. Rørdam,
Infinite non-simple
[31]
E. Kirchberg and M. Rørdam,
Purely infinite
[32] E. Kirchberg and A. Sierakowski, Strong pure infiniteness of crossed products, Ergodic Theory Dynam. Systems 38 (2018), no. 1, 220–243. 10.1017/etds.2016.25Search in Google Scholar
[33]
B. K. Kwaśniewski and W. Szymański,
Pure infiniteness and ideal structure of
[34] H. Lin, Stable approximate unitary equivalence of homomorphisms, J. Operator Theory 47 (2002), no. 2, 343–378. Search in Google Scholar
[35]
H. Matui and Y. Sato,
Decomposition rank of UHF-absorbing
[36] E. Michael, Continuous selections. I, Ann. of Math. (2) 63 (1956), 361–382. 10.1016/B978-044450355-8/50029-5Search in Google Scholar
[37] E. Michael, A selection theorem, Proc. Amer. Math. Soc. 17 (1966), 1404–1406. 10.1090/S0002-9939-1966-0203702-5Search in Google Scholar
[38]
G. K. Pedersen,
[39]
N. C. Phillips,
A classification theorem for nuclear purely infinite simple
[40]
M. Rørdam,
A short proof of Elliott’s theorem:
[41]
M. Rørdam,
Classification of certain infinite simple
[42]
M. Rørdam,
Classification of nuclear, simple
[43] J. Rosenberg and C. Schochet, The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized K-functor, Duke Math. J. 55 (1987), no. 2, 431–474. 10.1215/S0012-7094-87-05524-4Search in Google Scholar
[44]
A. Tikuisis, S. White and W. Winter,
Quasidiagonality of nuclear
[45]
A. S. Toms and W. Winter,
Strongly self-absorbing
[46] W. Winter and J. Zacharias, Completely positive maps of order zero, Münster J. Math. 2 (2009), 311–324. Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston