Comparison and pure infiniteness of crossed products
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Abstract:
Let $\alpha : G\curvearrowright X$ be a continuous action of an infinite countable group on a compact Hausdorff space. We show that, under the hypothesis that the action $\alpha$ is topologically free and has no $G$-invariant regular Borel probability measure on $X$, dynamical comparison implies that the reduced crossed product of $\alpha$ is purely infinite and simple. This result, as an application, shows a dichotomy between stable finiteness and pure infiniteness for reduced crossed products arising from actions satisfying dynamical comparison. We also introduce the concepts of paradoxical comparison and the uniform tower property. Under the hypothesis that the action $\alpha$ is exact and essentially free, we show that paradoxical comparison together with the uniform tower property implies that the reduced crossed product of $\alpha$ is purely infinite. As applications, we provide new results on pure infiniteness of reduced crossed products in which the underlying spaces are not necessarily zero dimensional. Finally, we study the type semigroups of actions on the Cantor set in order to establish the equivalence of almost unperforation of the type semigroup and comparison. This sheds light on a question arising in a paper of Rørdam and Sierakowski.References
- Pere Ara and Ruy Exel, Dynamical systems associated to separated graphs, graph algebras, and paradoxical decompositions, Adv. Math. 252 (2014), 748–804. MR 3144248, DOI 10.1016/j.aim.2013.11.009
- Pere Ara, Francesc Perera, and Andrew S. Toms, $K$-theory for operator algebras. Classification of $C^*$-algebras, Aspects of operator algebras and applications, Contemp. Math., vol. 534, Amer. Math. Soc., Providence, RI, 2011, pp. 1–71. MR 2767222, DOI 10.1090/conm/534/10521
- R. J. Archbold and J. S. Spielberg, Topologically free actions and ideals in discrete $C^*$-dynamical systems, Proc. Edinburgh Math. Soc. (2) 37 (1994), no. 1, 119–124. MR 1258035, DOI 10.1017/S0013091500018733
- Bruce E. Blackadar and Joachim Cuntz, The structure of stable algebraically simple $C^{\ast }$-algebras, Amer. J. Math. 104 (1982), no. 4, 813–822. MR 667536, DOI 10.2307/2374206
- Nathanial P. Brown and Narutaka Ozawa, $C^*$-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008. MR 2391387, DOI 10.1090/gsm/088
- J. Buck, Smallness and comparison properties for minimal dynamical systems, arXiv:1306.6681v1 (2013).
- T. Hines, The radius of comparison and mean dimension, ProQuest LLC, Ann Arbor, MI, 2015. Thesis (Ph.D.)–Purdue University.
- D. Kerr, Dimension, comparison, and almost finiteness, arXiv:1710.00393 (2017).
- D. Kerr and G. Szabó, Almost finiteness and the small boundary property, arXiv:1807.04326 (2018).
- Eberhard Kirchberg and Mikael Rørdam, Non-simple purely infinite $C^\ast$-algebras, Amer. J. Math. 122 (2000), no. 3, 637–666. MR 1759891, DOI 10.1353/ajm.2000.0021
- Eberhard Kirchberg and Mikael Rørdam, Infinite non-simple $C^*$-algebras: absorbing the Cuntz algebras $\scr O_\infty$, Adv. Math. 167 (2002), no. 2, 195–264. MR 1906257, DOI 10.1006/aima.2001.2041
- E. Kirchberg and A. Sierakowski, Filling families and strong pure infiniteness, arXiv:1503.08519 (2015).
- Marcelo Laca and Jack Spielberg, Purely infinite $C^*$-algebras from boundary actions of discrete groups, J. Reine Angew. Math. 480 (1996), 125–139. MR 1420560, DOI 10.1515/crll.1996.480.125
- Paul Jolissaint and Guyan Robertson, Simple purely infinite $C^\ast$-algebras and $n$-filling actions, J. Funct. Anal. 175 (2000), no. 1, 197–213. MR 1774856, DOI 10.1006/jfan.2000.3608
- Eduard Ortega, Francesc Perera, and Mikael Rørdam, The corona factorization property, stability, and the Cuntz semigroup of a $C^\ast$-algebra, Int. Math. Res. Not. IMRN 1 (2012), 34–66. MR 2874927, DOI 10.1093/imrn/rnr013
- N. C. Philips, Large subalgebras, arXiv:1408:5546 (2014).
- Timothy Rainone, Finiteness and paradoxical decompositions in $\rm C^*$-dynamical systems, J. Noncommut. Geom. 11 (2017), no. 2, 791–822. MR 3669119, DOI 10.4171/JNCG/11-2-11
- Mikael Rørdam, A simple $C^*$-algebra with a finite and an infinite projection, Acta Math. 191 (2003), no. 1, 109–142. MR 2020420, DOI 10.1007/BF02392697
- M. Rørdam, Classification of nuclear, simple $C^*$-algebras, Classification of nuclear $C^*$-algebras. Entropy in operator algebras, Encyclopaedia Math. Sci., vol. 126, Springer, Berlin, 2002, pp. 1–145. MR 1878882, DOI 10.1007/978-3-662-04825-2_{1}
- Mikael Rørdam, The stable and the real rank of $\scr Z$-absorbing $C^*$-algebras, Internat. J. Math. 15 (2004), no. 10, 1065–1084. MR 2106263, DOI 10.1142/S0129167X04002661
- Mikael Rørdam and Adam Sierakowski, Purely infinite $C^*$-algebras arising from crossed products, Ergodic Theory Dynam. Systems 32 (2012), no. 1, 273–293. MR 2873171, DOI 10.1017/S0143385710000829
- Jean Renault, The ideal structure of groupoid crossed product $C^\ast$-algebras, J. Operator Theory 25 (1991), no. 1, 3–36. With an appendix by Georges Skandalis. MR 1191252
- Adam Sierakowski, The ideal structure of reduced crossed products, Münster J. Math. 3 (2010), 237–261. MR 2775364
- Stan Wagon, The Banach-Tarski paradox, Cambridge University Press, Cambridge, 1993. With a foreword by Jan Mycielski; Corrected reprint of the 1985 original. MR 1251963
- Dana P. Williams, Crossed products of $C{^\ast }$-algebras, Mathematical Surveys and Monographs, vol. 134, American Mathematical Society, Providence, RI, 2007. MR 2288954, DOI 10.1090/surv/134
Additional Information
- Xin Ma
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
- MR Author ID: 1256218
- Email: dongodel@math.tamu.edu
- Received by editor(s): September 12, 2018
- Received by editor(s) in revised form: June 3, 2019
- Published electronically: August 28, 2019
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 7497-7520
- MSC (2010): Primary 37B05, 46L35
- DOI: https://doi.org/10.1090/tran/7927
- MathSciNet review: 4024559