Illinois Journal of Mathematics

Group actions on labeled graphs and their $C^{*}$-algebras

Teresa Bates, David Pask, and Paulette Willis

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Abstract

We introduce the notion of the action of a group on a labeled graph and the quotient object, also a labeled graph. We define a skew product labeled graph and use it to prove a version of the Gross–Tucker theorem for labeled graphs. We then apply these results to the $C^{*}$-algebra associated to a labeled graph and provide some applications in non-Abelian duality.

Article information

Source
Illinois J. Math., Volume 56, Number 4 (2012), 1149-1168.

Dates
First available in Project Euclid: 6 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1399395826

Digital Object Identifier
doi:10.1215/ijm/1399395826

Mathematical Reviews number (MathSciNet)
MR3231477

Zentralblatt MATH identifier
1292.46034

Subjects
Primary: 46L05: General theory of $C^*$-algebras
Secondary: 37B10: Symbolic dynamics [See also 37Cxx, 37Dxx]

Citation

Bates, Teresa; Pask, David; Willis, Paulette. Group actions on labeled graphs and their $C^{*}$-algebras. Illinois J. Math. 56 (2012), no. 4, 1149--1168. doi:10.1215/ijm/1399395826. https://projecteuclid.org/euclid.ijm/1399395826


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References

  • T. Bates, T. Carlsen and D. Pask, $C^*$-algebras of labelled graphs III–-K-theory, available at \arxivurlarXiv:1203.3072v1 [math.OA].
  • T. Bates and D. Pask, $C^*$-algebras of labeled graphs, J. Operator Theory 57 (2007), 207–226.
  • T. Bates and D. Pask, $C^*$-algebras of labeled graphs II–-Simplicity results, Math. Scand. 104 (2009), 249–274.
  • T. Crisp, Corners of graph algebras, J. Operator Theory 60 (2008), 253–271.
  • K. Deicke, D. Pask and I. Raeburn, Coverings of directed graphs and crossed products of $C^*$-algebras by coactions of homogenous spaces, Internat. J. Math. 14 (2003), 773–789.
  • J. Gross and T. Tucker, Topological graph theory, Series in Discrete Mathematics and Optimization, Wiley, New York, 1987.
  • J. Jeong and S. Kim, On simple labelled graph $C^*$-algebras, J. Math. Anal. Appl. 386 (2012), 631–640.
  • J. Jeong, S. Kim and G. Park, The structure of gauge invariant ideals of labelled graph $C^*$-algebras, J. Funct. Anal. 262 (2012), 1759–1780.
  • S. Kaliszewski, J. Quigg and I. Raeburn, Skew products and crossed products by coactions, J. Operator Theory 46 (2001), 411–433.
  • Y. Katayama, Takesaki's duality for a non-degenerate co-action, Math. Scand. 55 (1985), 141–151.
  • A. Kumjian and D. Pask, $C^*$-algebras of directed graphs and group actions, Ergodic Theory Dynam. Systems 19 (1999), 1503–1519.
  • D. Lind and B. Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995.
  • D. Pask, J. Quigg and I. Raeburn, Coverings of $k$-graphs, J. Algebra 289 (2005), 161–191.
  • D. Pask and I. Raeburn, On the $K$-theory of Cuntz–Krieger algebras, Publ. RIMS Kyoto 32 (1996), 415–443.
  • J. Quigg, Discrete $C^*$-coactions and $C^*$-algebraic bundles, J. Austral. Math. Soc. Ser. A 60 (1996), 204–221.
  • D. Robertson and W. Szymański, $C^*$-algebras associated to $C^*$-correspondences and applications to mirror quantum spheres, Illinois J. Math. 55 (2011), 845–870.