## Abstract and Applied Analysis

### The Tracial Class Property for Crossed Products by Finite Group Actions

#### Abstract

We define the concept of tracial $\mathrm{\scr C}$-algebra of ${C}^{\ast}$-algebras, which generalize the concept of local $\mathrm{\scr C}$-algebra of ${C}^{\ast}$-algebras given by H. Osaka and N. C. Phillips. Let $\mathrm{\scr C}$ be any class of separable unital ${C}^{\ast}$-algebras. Let $A$ be an infinite dimensional simple unital tracial $\mathrm{\scr C}$-algebra with the (SP)-property, and let $\alpha :G\to \text{A}\text{u}\text{t}(A)$ be an action of a finite group $G$ on $A$ which has the tracial Rokhlin property. Then $\mathrm{A }{{\times}}_{\alpha }\mathrm{ G}$ is a simple unital tracial $\mathrm{\scr C}$-algebra.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 745369, 10 pages.

Dates
First available in Project Euclid: 28 March 2013

https://projecteuclid.org/euclid.aaa/1364475915

Digital Object Identifier
doi:10.1155/2012/745369

Mathematical Reviews number (MathSciNet)
MR2994939

Zentralblatt MATH identifier
1262.46046

#### Citation

Yang, Xinbing; Fang, Xiaochun. The Tracial Class Property for Crossed Products by Finite Group Actions. Abstr. Appl. Anal. 2012 (2012), Article ID 745369, 10 pages. doi:10.1155/2012/745369. https://projecteuclid.org/euclid.aaa/1364475915

#### References

• H. Lin, “The tracial topological rank of ${C}^{\ast\,\!}$-algebras,” Proceedings of the London Mathematical Society, vol. 83, no. 1, pp. 199–234, 2001.
• H. Lin, “Classification of simple ${C}^{\ast\,\!}$-algebras of tracial topological rank zero,” Duke Mathematical Journal, vol. 125, no. 1, pp. 91–119, 2004.
• H. Lin, “Classification of simple ${C}^{\ast\,\!}$-algebras and higher dimensional noncommutative tori,” Annals of Mathematics, vol. 157, no. 2, pp. 521–544, 2003.
• X. Fang, “The classification of certain non-simple ${C}^{\ast\,\!}$-algebras of tracial rank zero,” Journal of Functional Analysis, vol. 256, no. 12, pp. 3861–3891, 2009.
• H. Yao and S. Hu, “${C}^{\ast\,\!}$-algebras of tracial real rank zero,” Journal of East China Normal University, vol. 2, pp. 5–12, 2004.
• Q. Z. Fan and X. C. Fang, “${C}^{\ast\,\!}$-algebras of tracially stable rank one,” Acta Mathematica Sinica, vol. 48, no. 5, pp. 929–934, 2005.
• G. A. Elliott and Z. Niu, “On tracial approximation,” Journal of Functional Analysis, vol. 254, no. 2, pp. 396–440, 2008.
• X. Fang and Q. Fan, “čommentComment on ref. [6?]: Please update the information of this reference, if possible.Certain properties for crossed products by automorphisms with a certain non-simple tracial Rokhlin property,” Ergodic Theory and Dynamical Systems. In press.
• A. Connes, “Outer conjugacy classes of automorphisms of factors,” Annales Scientifiques de l'École Normale Supérieure, vol. 8, no. 3, pp. 383–419, 1975.
• R. H. Herman and A. Ocneanu, “Stability for integer actions on UHF ${C}^{\ast\,\!}$-algebras,” Journal of Functional Analysis, vol. 59, no. 1, pp. 132–144, 1984.
• M. Rørdam, “Classification of certain infinite simple ${C}^{\ast\,\!}$-algebras,” Journal of Functional Analysis, vol. 131, no. 2, pp. 415–458, 1995.
• A. Kishimoto, “The Rohlin property for shifts on UHF algebras and automorphisms of Cuntz algebras,” Journal of Functional Analysis, vol. 140, no. 1, pp. 100–123, 1996.
• N. C. Phillips, “The tracial Rokhlin property for actions of finite groups on ${C}^{\ast\,\!}$-algebras,” American Journal of Mathematics, vol. 133, no. 3, pp. 581–636, 2011.
• H. Osaka and N. C. Phillips, “Crossed products by finite group actions with the Rokhlin property,” Mathematische Zeitschrift, vol. 270, no. 1-2, pp. 19–42, 2012.
• X. Fang, “The real rank zero property of crossed product,” Proceedings of the American Mathematical Society, vol. 134, no. 10, pp. 3015–3024, 2006.
• H. Lin and H. Osaka, “The Rokhlin property and the tracial topological rank,” Journal of Functional Analysis, vol. 218, no. 2, pp. 475–494, 2005.
• H. Osaka and N. C. Phillips, “Stable and real rank for crossed products by automorphisms with the tracial Rokhlin property,” Ergodic Theory and Dynamical Systems, vol. 26, no. 5, pp. 1579–1621, 2006.
• X. Yang and X. Fang, “The tracial rank for crossed products by finite group actions,” The Rocky Mountain Journal of Mathematics, vol. 42, no. 1, pp. 339–352, 2012.
• H. Lin, An Introduction to the Classification of Amenable C$^{\ast\,\!}$-Algebras, World Scientific Publishing, River Edge, NJ, USA, 2001.
• J. A. Jeong and H. Osaka, “Extremally rich ${C}^{\ast\,\!}$-crossed products and the cancellation property,” Australian Mathematical Society A, vol. 64, no. 3, pp. 285–301, 1998.