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X-ROKHLIN PROPERTY FOR ACTIONS OF COUNTABLE DISCRETE GROUPS

Part of: Selfadjoint operator algebras Linear spaces and algebras of operators

Published online by Cambridge University Press:  04 July 2025

XIAOCHUN FANG Open the ORCID record for XIAOCHUN FANG [Opens in a new window]  and
XIN CAO Open the ORCID record for XIN CAO [Opens in a new window]
XIAOCHUN FANG
Affiliation:
School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Ministry of Education), https://ror.org/03rc6as71Tongji University, Shanghai 200092, PR China e-mail: xfang@tongji.edu.cn
XIN CAO*
Affiliation:
School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Ministry of Education), https://ror.org/03rc6as71Tongji University, Shanghai 200092, PR China
*
e-mail: 2111153@tongji.edu.cn
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Abstract

We extend the definition of the X-Rokhlin property to countable discrete groups and prove some permanence properties. If the action of a countable discrete group on X is free and minimal and the action of this group on the separable simple $C^*$-algebra has the X-Rokhlin property, then the reduced crossed product is simple.

Keywords

C*-algebracrossed productRokhlin property

MSC classification

Primary: 47L65: Crossed product algebras (analytic crossed products)
Secondary: 46L05: General theory of $C^*$-algebras

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 11
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc

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References

Amini, M., Golestani, N., Jamali, S. and Phillips, N. C., ‘Finite group and integer actions on simple tracially $\mathcal{Z}$ -absorbing ${C}^{\ast }$ -algebras’, J. Operator Theory 92(2) (2024), 505–547.10.7900/jot.2022nov02.2417CrossRefGoogle Scholar
Choi, M. D. and Effros, E. G., ‘The completely positive lifting problem for ${C}^{\ast }$ -algebras’, Ann. of Math. (2) 104(3) (1976), 585–609.10.2307/1970968CrossRefGoogle Scholar
Gardella, E., ‘Rokhlin dimension for compact group actions’, Indiana Univ. Math. J. 66(2) (2017), 659–703.10.1512/iumj.2017.66.5951CrossRefGoogle Scholar
Gardella, E., ‘Regularity properties and Rokhlin dimension for compact group actions’, Houston J. Math. 43(3) (2017), 861–889.Google Scholar
Hirshberg, I. and Phillips, N. C., ‘Rokhlin dimension: obstructions and permanence properties’, Doc. Math. 20 (2015), 199–236.10.4171/dm/489CrossRefGoogle Scholar
Hirshberg, I., Szabó, G., Winter, W. and Wu, J., ‘Rokhlin dimension for flows’, Comm. Math. Phys. 353(1) (2017), 253–316.10.1007/s00220-016-2762-0CrossRefGoogle Scholar
Hirshberg, I., Winter, W. and Zacharias, J., ‘Rokhlin dimension and ${C}^{\ast }$ -dynamics’, Comm. Math. Phys. 335(2) (2015), 637–670.10.1007/s00220-014-2264-xCrossRefGoogle Scholar
Izumi, M., ‘Finite group actions on ${C}^{\ast }$ -algebras with the Rokhlin property. I’, Duke Math. J. 122(2) (2004), 233–280.10.1215/S0012-7094-04-12221-3CrossRefGoogle Scholar
Izumi, M., ‘Finite group actions on ${C}^{\ast }$ -algebras with the Rokhlin property. II’, Adv. Math. 184(1) (2004), 119–160.10.1016/S0001-8708(03)00140-3CrossRefGoogle Scholar
Kishimoto, A., ‘Outer automorphisms and reduced crossed products of simple ${C}^{\ast }$ -algebras’, Comm. Math. Phys. 81(3) (1981), 429–435.10.1007/BF01209077CrossRefGoogle Scholar
Phillips, N. C., ‘The tracial Rokhlin property for actions of finite groups on ${C}^{\ast }$ -algebras’, Amer. J. Math. 133(3) (2011), 581–636.10.1353/ajm.2011.0016CrossRefGoogle Scholar
Sureshkumar, M. and Vaidyanathan, P., ‘Rokhlin dimension: permanence properties and ideal separation’, Groups Geom. Dyn., to appear; doi:10.4171/GGD/825.CrossRefGoogle Scholar
Szabó, G., Wu, J. and Zacharias, J., ‘Rokhlin dimension for actions of residually finite groups’, Ergodic Theory Dynam. Systems 39(8) (2019), 2248–2304.10.1017/etds.2017.113CrossRefGoogle Scholar