Abstract and Applied Analysis

${C}^{\ast }$-Algebras from Groupoids on Self-Similar Groups

Inhyeop Yi

Abstract

We show that the Smale spaces from self-similar groups are topologically mixing and their stable algebra and stable Ruelle algebra are strongly Morita equivalent to groupoid algebras of Anantharaman-Delaroche and Deaconu. And we show that ${C}^{\ast }({R}_{\infty })$ associated to a postcritically finite hyperbolic rational function is an AT-algebra of real-rank zero with a unique trace state.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 579042, 7 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393450165

Digital Object Identifier
doi:10.1155/2013/579042

Mathematical Reviews number (MathSciNet)
MR3093769

Zentralblatt MATH identifier
1300.46051

Citation

Yi, Inhyeop. ${C}^{\ast }$ -Algebras from Groupoids on Self-Similar Groups. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 579042, 7 pages. doi:10.1155/2013/579042. https://projecteuclid.org/euclid.aaa/1393450165

References

• V. Nekrashevych, Self-Similar Groups, vol. 117 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 2005.
• V. Nekrashevych, “${C}^{\ast\,\!}$-algebras and self-similar groups,” Journal für die Reine und Angewandte Mathematik, vol. 630, pp. 59–123, 2009.
• I. F. Putnam, “${C}^{\ast\,\!}$-algebras from Smale spaces,” Canadian Journal of Mathematics, vol. 48, no. 1, pp. 175–195, 1996.
• I. Putnam, Hyperbolic Systems and Generalized Cuntz-Krieger Algebras, Lecture notes from Summer School in Operator Algebras, Odense, Denmark, 1996.
• I. F. Putnam and J. Spielberg, “The structure of ${C}^{\ast\,\!}$-algebras associated with hyperbolic dynamical systems,” Journal of Functional Analysis, vol. 163, no. 2, pp. 279–299, 1999.
• D. Ruelle, Thermodynamic Formalism, vol. 5, Addison-Wesley, 1978.
• D. Ruelle, “Noncommutative algebras for hyperbolic diffeomorphisms,” Inventiones Mathematicae, vol. 93, no. 1, pp. 1–13, 1988.
• C. Anantharaman-Delaroche, “Purely infinite ${C}^{\ast\,\!}$-algebras arising from dynamical systems,” Bulletin de la Société Mathématique de France, vol. 125, no. 2, pp. 199–225, 1997.
• V. Deaconu, “Groupoids associated with endomorphisms,” Transactions of the American Mathematical Society, vol. 347, no. 5, pp. 1779–1786, 1995.
• P. S. Muhly, J. N. Renault, and D. P. Williams, “Equivalence and isomorphism for groupoid ${C}^{\ast\,\!}$-algebras,” Journal of Operator Theory, vol. 17, no. 1, pp. 3–22, 1987.
• X. Chen and C. Hou, “Morita equivalence of groupoid ${C}^{\ast\,\!}$-algebras arising from dynamical systems,” Studia Mathematica, vol. 149, no. 2, pp. 121–132, 2002.
• J. Renault, “Cuntz-like algebras,” in Operator Theoretical Methods, pp. 371–386, Bucharest, Romania, Theta Foundation, 2000.
• A. Kumjian and D. Pask, “Actions of ${\mathbb{Z}}^{k}$ associated to higher rank graphs,” Ergodic Theory and Dynamical Systems, vol. 23, no. 4, pp. 1153–1172, 2003.
• G. A. Elliott, G. Gong, and L. Li, “Approximate divisibility of simple inductive limit ${C}^{\ast\,\!}$-algebras,” Contemporary Mathematics, vol. 228, pp. 87–97, 1998.
• J. Renault, A Groupoid Approach to C$^{\ast\,\!}$-Algebras, vol. 793 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1980.
• S. Zhang, “Certain ${C}^{\ast\,\!}$-algebras with real rank zero and their corona and multiplier algebras. I,” Pacific Journal of Mathematics, vol. 155, no. 1, pp. 169–197, 1992.
• J.-L. Tu, “La conjecture de Baum-Connes pour les feuilletages moyennables,” K-Theory in the Mathematical Sciences, vol. 17, no. 3, pp. 215–264, 1999.
• J. Renault, “The Radon-Nikodym problem for appoximately proper equivalence relations,” Ergodic Theory and Dynamical Systems, vol. 25, no. 5, pp. 1643–1672, 2005.
• A. Kumjian and J. Renault, “KMS states on ${C}^{\ast\,\!}$-algebras asso-ciated to expansive maps,” Proceedings of the American Mathematical Society, vol. 134, no. 7, pp. 2067–2078, 2006.
• P. S. Muhly and D. P. Williams, “Continuous trace groupoid ${C}^{\ast\,\!}$-algebras,” Mathematica Scandinavica, vol. 66, no. 2, pp. 231–241, 1990.
• A. An Huef, I. Raeburn, and D. P. Williams, “Properties preserved under Morita equivalence of ${C}^{\ast\,\!}$-algebras,” Proceedings of the American Mathematical Society, vol. 135, no. 5, pp. 1495–1503, 2007.
• K. Thomsen, “${C}^{\ast\,\!}$-algebras of homoclinic and heteroclinic structure in expansive dynamics,” Memoirs of the American Mathematical Society, vol. 206, no. 970, 2010.
• G. Gong, “On the classification of simple inductive limit ${C}^{\ast\,\!}$-algebras–-I. The reduction theorem,” Documenta Mathematica, vol. 7, pp. 255–461, 2002.
• B. Blackadar, A. Kumjian, and M. Rørdam, “Approximately central matrix units and the structure of noncommutative tori,” K-Theory, vol. 6, no. 3, pp. 267–284, 1992.