Abstract and Applied Analysis

Asymptotic C -Algebras from k -Actions on Higher Rank Graphs

Inhyeop Yi

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Abstract

For a dynamical system arising from k -action on a higher rank graph with finite vertex set, we show that the semidirect product of the asymptotic equivalence relation groupoid is essentially principal if and only if the k -graph satisfies the aperiodic condition. Then we show that the corresponding asymptotic Ruelle algebra is simple if the graph is primitive with the aperiodic condition.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 752497, 5 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393449619

Digital Object Identifier
doi:10.1155/2013/752497

Mathematical Reviews number (MathSciNet)
MR3068868

Citation

Yi, Inhyeop. Asymptotic ${C}^{\ast }$ -Algebras from ${\Bbb Z}^{k}$ -Actions on Higher Rank Graphs. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 752497, 5 pages. doi:10.1155/2013/752497. https://projecteuclid.org/euclid.aaa/1393449619


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References

  • 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.
  • I. F. Putnam, “${C}^{\ast\,\!}$-algebras from Smale spaces,” Canadian Journal of Mathematics, vol. 48, no. 1, pp. 175–195, 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 of Encyclopedia of Mathematics and Its Applications, Addison-Wesley, 1978.
  • D. Ruelle, “Noncommutative algebras for hyperbolic diffeomorphisms,” Inventiones Mathematicae, vol. 93, no. 1, pp. 1–13, 1988.
  • J. Cuntz and W. Krieger, “A class of ${C}^{\ast\,\!}$-algebras and topological Markov chains,” Inventiones Mathematicae, vol. 56, no. 3, pp. 251–268, 1980.
  • C. Hou and X. Chen, “A note on the ideals of groupoid ${C}^{\ast\,\!}$-algebras from Smale spaces,” Bulletin of the Australian Mathematical Society, vol. 64, no. 2, pp. 271–279, 2001.
  • A. Kumjian and D. Pask, “Higher rank graph ${C}^{\ast\,\!}$-algebras,” New York Journal of Mathematics, vol. 6, pp. 1–20, 2000.
  • J. Renault, A Groupoid Approach to C$^{\ast\,\!}$-Algebras, vol. 793 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1980.