Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-12-01T02:05:36.777Z Has data issue: false hasContentIssue false
  • English
  • Français

Equivariant $\mathcal {O}_{2}$-absorption theorem for exact groups

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  17 June 2021

Yuhei Suzuki
Yuhei Suzuki*
Affiliation:
Department of Mathematics, Faculty of Science, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido060-0810, [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that, up to strong cocycle conjugacy, every countable exact group admits a unique equivariantly $\mathcal {O}_{2}$-absorbing, pointwise outer action on the Cuntz algebra $\mathcal {O}_{2}$ with the quasi-central approximation property (QAP). In particular, we establish the equivariant analogue of the Kirchberg $\mathcal {O}_{2}$-absorption theorem for these groups.

Keywords

C*-dynamical systemsstrong cocycle conjugacyamenable actions

MSC classification

Primary: 46L55: Noncommutative dynamical systems
Secondary: 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 7 , July 2021 , pp. 1492 - 1506
Check for updates
Copyright
© 2021 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anantharaman-Delaroche, C., Action moyennable d'un groupe localement compact sur une algèbre de von Neumann, Math. Scand. 45 (1979), 289–304.10.7146/math.scand.a-11844CrossRefGoogle Scholar
Anantharaman-Delaroche, C., Systèmes dynamiques non commutatifs et moyennabilité, Math. Ann. 279 (1987), 297–315.CrossRefGoogle Scholar
Anantharaman-Delaroche, C., Amenability and exactness for dynamical systems and their $C^{\ast }$-algebras, Trans. Amer. Math. Soc. 354 (2002), 4153–4178.10.1090/S0002-9947-02-02978-1CrossRefGoogle Scholar
Brown, N. P. and Ozawa, N., $C^{\ast }$-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88 (American Mathematical Society, Providence, RI, 2008).Google Scholar
Buss, A., Echterhoff, S. and Willett, R., Injectivity, crossed products, and amenable group actions, Contemp. Math. 749 (2020), 105–138.10.1090/conm/749/15069CrossRefGoogle Scholar
Connes, A., Periodic automorphisms of the hyperfinite factors of type ${\rm II}_1$, Acta Sci. Math. (Szeged) 39 (1977), 39–66.Google Scholar
Cuntz, J., K-theory for certain $C^{\ast }$-algebras, Ann. of Math. (2) 113 (1981), 181–197.CrossRefGoogle Scholar
Farah, I. and Hart, B., Countable saturation of corona algebras, C. R. Math. Acad. Sci. Soc. R. Can. 35 (2013), 35–56.Google Scholar
Gromov, M., Random walk in random groups, Geom. Funct. Anal. 13 (2003), 73–146.CrossRefGoogle Scholar
Izumi, M., Finite group actions on $C^{\ast }$-algebras with the Rohlin property I, Duke Math. J. 122 (2004), 233–280.10.1215/S0012-7094-04-12221-3CrossRefGoogle Scholar
Izumi, M., Group actions on operator algebras, in Proceedings of the International Congress of Mathematicians 2010 (ICM 2010), Volume III (Hindustan Book Agency, 2011), 1528–1548.CrossRefGoogle Scholar
Izumi, M. and Matui, H., Poly-$\mathbb {Z}$ group actions on Kirchberg algebras I, Int. Math. Res. Not. IMRN, to appear. Preprint (2018), arXiv:1810.05850.Google Scholar
Izumi, M. and Matui, H., Poly-$\mathbb {Z}$ group actions on Kirchberg algebras II, Invent. Math., to appear. Preprint (2019), arXiv:1906.03818v2.Google Scholar
Kirchberg, E., The classification of purely infinite $C^{\ast }$-algebras using Kasparov's theory, Preprint.Google Scholar
Kirchberg, E. and Phillips, N. C., Embedding of exact $C^{\ast }$-algebras in the Cuntz algebra $\mathcal {O}_{2}$, J. Reine Angew. Math. 525 (2000), 17–53.CrossRefGoogle Scholar
Kirchberg, E. and Wassermann, S., Exact groups and continuous bundles of $C^{\ast }$-algebras, Math. Ann. 315 (1999), 169–203.10.1007/s002080050364CrossRefGoogle Scholar
Matui, H., Classification of outer actions of $\mathbb {Z}^N$ on $\mathcal {O}_{2}$, Adv. Math. 217 (2008), 2872–2896.CrossRefGoogle Scholar
Nakamura, H., Aperiodic automorphisms of nuclear purely infinite simple $C^{\ast }$-algebras, Ergodic Theory Dynam. Systems 20 (2000), 1749–1765.CrossRefGoogle Scholar
Osajda, D., Small cancellation labellings of some infinite graphs and applications, Acta Math. 225 (2020), 159–191.CrossRefGoogle Scholar
Ozawa, N., Amenable actions and exactness for discrete groups, C. R. Acad. Sci. Paris Ser. I Math. 330 (2000), 691–695.CrossRefGoogle Scholar
Ozawa, N., Amenable actions and applications, in Proceedings of the International Congress of Mathematicians, Madrid 2006, Vol. II (European Mathematical Society, Zürich, 2007), 1563–1580.Google Scholar
Phillips, N. C., A classification theorem for nuclear purely infinite simple ${C}^{\ast }$-algebras, Doc. Math. 5 (2000), 49–114.Google Scholar
Rørdam, M., Classification of nuclear, simple ${C}^{\ast }$-algebras, Encyclopaedia of Mathematical Sciences, vol. 126 (Springer, Berlin, 2002), 1–145.Google Scholar
Suzuki, Y., Simple equivariant $C^{\ast }$-algebras whose full and reduced crossed products coincide, J. Noncommut. Geom. 13 (2019), 1577–1585.10.4171/JNCG/356CrossRefGoogle Scholar
Suzuki, Y., Non-amenable tight squeezes by Kirchberg algebras, Preprint (2019), arXiv:1908.02971.Google Scholar
Suzuki, Y., Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems, Comm. Math. Phys. 375 (2020), 1273–1297.10.1007/s00220-019-03436-1CrossRefGoogle Scholar
Suzuki, Y., On pathological properties of fixed point algebras in Kirchberg algebras, Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), 3087–3096.10.1017/prm.2019.47CrossRefGoogle Scholar
Szabó, G., Strongly self-absorbing $C^{\ast }$-dynamical systems, Trans. Amer. Math. Soc. 370 (2018), 99–130 (with an erratum).CrossRefGoogle Scholar
Szabó, G., Strongly self-absorbing $C^{\ast }$-dynamical systems II, J. Noncommut. Geom. 12 (2018), 369–406.10.4171/JNCG/279CrossRefGoogle Scholar
Szabó, G., Equivariant Kirchberg-Phillips-type absorption for amenable group actions, Comm. Math. Phys. 361 (2018), 1115–1154.10.1007/s00220-018-3110-3CrossRefGoogle Scholar
Szabó, G., Actions of certain torsion-free elementary amenable groups on strongly self-absorbing $C^{\ast }$-algebras, Comm. Math. Phys. 371 (2019), 267–284.10.1007/s00220-019-03435-2CrossRefGoogle Scholar
Toms, A. S. and Winter, W., Strongly self-absorbing $C^{\ast }$-algebras, Trans. Amer. Math. Soc. 359 (2007), 3999–4029.CrossRefGoogle Scholar