Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-01-12T04:17:25.776Z Has data issue: false hasContentIssue false
  • English
  • Français

Isometric actions and finite approximations

Part of: Topological dynamics Linear spaces and algebras of operators Special aspects of infinite or finite groups

Published online by Cambridge University Press:  06 June 2022

S. J. PILGRIM Open the ORCID record for S. J. PILGRIM [Opens in a new window]
S. J. PILGRIM*
Affiliation:
Department of Mathematics, The University of Hawai’i at Mānoa, Honolulu, USA
*
e-mail: spilgrim@hawaii.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that every isometric action on a Cantor set is conjugate to an inverse limit of actions on finite sets; and that every faithful isometric action by a finitely generated amenable group is residually finite.

Keywords

group actionsoperator algebrasquasidiagonalresidually finite

MSC classification

Primary: 37B99: None of the above, but in this section
Secondary: 47L65: Crossed product algebras (analytic crossed products) 20F65: Geometric group theory
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 7 , July 2023 , pp. 2464 - 2470
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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

de Cornulier, Y.. Groupes pleins-topologiques (d’après Matui, Juschenko, Monod,…). Astérisque 361(viii) (2014), 183–223, Exp. No. 1064.Google Scholar
Dixmier, J.. ${C}^{\ast }$ -algebras. North-Holland, Amsterdam, 1982.Google Scholar
Eilers, S., Shulman, T. and Sørensen, A. P. W.. C*-stability of discrete groups. Adv. Math. 373 (2020), 107324.10.1016/j.aim.2020.107324CrossRefGoogle Scholar
Grigorchuk, R. I., Nekrashevich, V. V. and Sushchanskii, V. I.. Automata, dynamical systems, and groups. Proc. Steklov Inst. Math. 231 (2000), 128–203.Google Scholar
Kerr, D. and Nowak, P.. Residually finite actions and crossed products. Ergod. Th. & Dynam. Sys. 32 (2012), 1585–1614.10.1017/S0143385711000575CrossRefGoogle Scholar
Nekrashevych, V.. Palindromic subshifts and simple periodic groups of intermediate growth. Ann. of Math. (2) 187(3) (2018), 667–719.10.4007/annals.2018.187.3.2CrossRefGoogle Scholar
Orfanos, S.. Generalized Bunce–Deddens algebras. Proc. Amer. Math. Soc. 138(1) (2010), 299–308.10.1090/S0002-9939-09-10071-0CrossRefGoogle Scholar
Tikuisis, A., White, S. and Winter, W.. Quasidiagonality of nuclear ${C}^{\ast }$ -algebras. Ann. of Math. (2) 185 (2017), 229–284.10.4007/annals.2017.185.1.4CrossRefGoogle Scholar
Yuan, Q.. Response to stack exchange question. https://math.stackexchange.com/questions/3854592/what-finitely-generated-amenable-groups-arise-as-subgroups-of-compact-lie-groups/3854768#3854768. Accessed October 15 2020.Google Scholar