A generalized Powers averaging property for commutative crossed products
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- by Tattwamasi Amrutam and Dan Ursu PDF
- Trans. Amer. Math. Soc. 375 (2022), 2237-2254 Request permission
Abstract:
We prove a generalized version of Powers’ averaging property that characterizes simplicity of reduced crossed products $C(X) \rtimes _\lambda G$, where $G$ is a countable discrete group, and $X$ is a compact Hausdorff space which $G$ acts on minimally by homeomorphisms. As a consequence, we generalize results of Hartman and Kalantar on unique stationarity to the state space of $C(X) \rtimes _\lambda G$ and to Kawabe’s generalized space of amenable subgroups $\operatorname {Sub}_a(X,G)$. This further lets us generalize a result of the first named author and Kalantar on simplicity of intermediate C*-algebras. We prove that if $C(Y) \subseteq C(X)$ is an inclusion of unital commutative $G$-C*-algebras with $X$ minimal and $C(Y) \rtimes _\lambda G$ simple, then any intermediate C*-algebra $A$ satisfying $C(Y) \rtimes _\lambda G \subseteq A \subseteq C(X) \rtimes _\lambda G$ is simple.References
- Tattwamasi Amrutam and Mehrdad Kalantar, On simplicity of intermediate $C^*$-algebras, Ergodic Theory Dynam. Systems 40 (2020), no. 12, 3181–3187. MR 4170599, DOI 10.1017/etds.2019.34
- Rasmus Sylvester Bryder and Matthew Kennedy, Reduced twisted crossed products over $\textrm {C}^*$-simple groups, Int. Math. Res. Not. IMRN 6 (2018), 1638–1655. MR 3801472, DOI 10.1093/imrn/rnw296
- Emmanuel Breuillard, Mehrdad Kalantar, Matthew Kennedy, and Narutaka Ozawa, $C^*$-simplicity and the unique trace property for discrete groups, Publ. Math. Inst. Hautes Études Sci. 126 (2017), 35–71. MR 3735864, DOI 10.1007/s10240-017-0091-2
- Harry Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math. (2) 77 (1963), 335–386. MR 146298, DOI 10.2307/1970220
- Harry Furstenberg, Boundary theory and stochastic processes on homogeneous spaces, Harmonic Analysis on Homogeneous Spaces (Calvin C. Moore, ed.), Proc. Sympos. Pure Math., vol. 26, Amer. Math. Soc., Providence, RI, 1973, pp. 193–229.
- Uffe Haagerup, A new look at $C^*$-simplicity and the unique trace property of a group, Operator algebras and applications—the Abel Symposium 2015, Abel Symp., vol. 12, Springer, [Cham], 2017, pp. 167–176. MR 3837596
- Yair Hartman and Mehrdad Kalantar, Stationary C*-dynamical systems, arXiv:1712.10133, 2017, to appear in J. Eur. Math. Soc. (JEMS).
- Takuya Kawabe, Uniformly recurrent subgroups and the ideal structure of reduced crossed products, arXiv:1701.03413, 2017.
- Matthew Kennedy, An intrinsic characterization of $C^*$-simplicity, Ann. Sci. Éc. Norm. Supér. (4) 53 (2020), no. 5, 1105–1119 (English, with English and French summaries). MR 4174855, DOI 10.24033/asens.2441
- Mehrdad Kalantar and Matthew Kennedy, Boundaries of reduced $C^*$-algebras of discrete groups, J. Reine Angew. Math. 727 (2017), 247–267. MR 3652252, DOI 10.1515/crelle-2014-0111
- Matthew Kennedy and Christopher Schafhauser, Noncommutative boundaries and the ideal structure of reduced crossed products, Duke Math. J. 168 (2019), no. 17, 3215–3260. MR 4030364, DOI 10.1215/00127094-2019-0032
- Zahra Naghavi, Furstenberg boundary of minimal actions, Integral Equations Operator Theory 92 (2020), no. 2, Paper No. 14, 14. MR 4078148, DOI 10.1007/s00020-020-2567-6
Additional Information
- Tattwamasi Amrutam
- Affiliation: Department of Mathematics, University of Houston, 3551 Cullen Blvd., Room 641, Philip Guthrie Hoffman Hall, Houston, Texas 77204-3008
- Address at time of publication: Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, Be’er Sheva 8410501, Israel
- MR Author ID: 1407185
- ORCID: 0000-0002-0691-5821
- Email: tattwama@post.bgu.ac.il
- Dan Ursu
- Affiliation: Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada
- ORCID: 0000-0002-3235-4589
- Email: dursu@uwaterloo.ca
- Received by editor(s): February 21, 2021
- Received by editor(s) in revised form: June 28, 2021, and September 29, 2021
- Published electronically: December 22, 2021
- Additional Notes: The second author was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) [grant number PGSD3-535032-2019]. Deuxième auteur financé par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG) [numéro de subvention PGSD3-535032-2019]
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 2237-2254
- MSC (2020): Primary 37A55, 46L55, 46L89, 47L65
- DOI: https://doi.org/10.1090/tran/8567
- MathSciNet review: 4378093