Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 10 (2014), 068, 14 pages      arXiv:1311.3880      https://doi.org/10.3842/SIGMA.2014.068
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

Groupoid Actions on Fractafolds

Marius Ionescu a and Alex Kumjian b
a) Department of Mathematics, United States Naval Academy, Annapolis, MD, 21402-5002, USA
b) Department of Mathematics, University of Nevada, Reno, NV, 89557, USA

Received February 04, 2014, in final form June 21, 2014; Published online June 28, 2014

Abstract
We define a bundle over a totally disconnected set such that each fiber is homeomorphic to a fractal blowup. We prove that there is a natural action of a Renault-Deaconu groupoid on our fractafold bundle and that the resulting action groupoid is a Renault-Deaconu groupoid itself. We also show that when the bundle is locally compact the associated $C^*$-algebra is primitive and has a densely defined lower-semicontinuous trace.

Key words: Renault-Deaconu groupoids; fractafolds; iterated function systems.

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References

  1. Anantharaman-Delaroche C., Purely infinite $C^*$-algebras arising from dynamical systems, Bull. Soc. Math. France 125 (1997), 199-225.
  2. Anantharaman-Delaroche C., Renault J., Amenable groupoids, in Groupoids in Analysis, Geometry, and Physics (Boulder, CO, 1999), Contemp. Math., Vol. 282, Amer. Math. Soc., Providence, RI, 2001, 35-46.
  3. Barnsley M.F., Fractals everywhere, 2nd ed., Academic Press Professional, Boston, MA, 1993.
  4. Blackadar B., Operator algebras. Theory of $C^{*}$-algebras and von Neumann algebras, operator algebras and non-commutative geometry, III, Encyclopaedia of Mathematical Sciences, Vol. 122, Springer-Verlag, Berlin, 2006.
  5. Brown J., Clark L.O., Farthing C., Sims A., Simplicity of algebras associated to étale groupoids, Semigroup Forum 88 (2014), 433-452, arXiv:1204.3127.
  6. Curtis D.W., Patching D.S., Hyperspaces of direct limits of locally compact metric spaces, Topology Appl. 29 (1988), 55-60.
  7. Deaconu V., Groupoids associated with endomorphisms, Trans. Amer. Math. Soc. 347 (1995), 1779-1786.
  8. Deaconu V., Kumjian A., Muhly P., Cohomology of topological graphs and Cuntz-Pimsner algebras, J. Operator Theory 46 (2001), 251-264, math.OA/9901094.
  9. Edgar G., Measure, topology, and fractal geometry, 2nd ed., Undergraduate Texts in Mathematics, Springer, New York, 2008.
  10. Exel R., Non-Hausdorff étale groupoids, Proc. Amer. Math. Soc. 139 (2011), 897-907, arXiv:0812.4087.
  11. Hutchinson J.E., Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747.
  12. McDonald J.N., Weiss N.A., A course in real analysis, 2nd ed., Academic Press Inc., Amsterdam, 2013.
  13. Muhly P.S., Renault J.N., Williams D.P., Equivalence and isomorphism for groupoid $C^\ast$-algebras, J. Operator Theory 17 (1987), 3-22.
  14. Pedersen G.K., $C^{\ast} $-algebras and their automorphism groups, London Mathematical Society Monographs, Vol. 14, Academic Press, Inc., London - New York, 1979.
  15. Pimsner M.V., A class of $C^*$-algebras generalizing both Cuntz-Krieger algebras and crossed products by ${\bf Z}$, in Free Probability Theory (Waterloo, ON, 1995), Fields Inst. Commun., Vol. 12, Amer. Math. Soc., Providence, RI, 1997, 189-212.
  16. Renault J., A groupoid approach to $C^{\ast}$-algebras, Lecture Notes in Mathematics, Vol. 793, Springer, Berlin, 1980.
  17. Renault J., Cuntz-like algebras, in Operator Theoretical Methods (Timişoara, 1998), Theta Found., Bucharest, 2000, 371-386, math.OA/9905185.
  18. Renault J., Cartan subalgebras in $C^*$-algebras, Irish Math. Soc. Bull. (2008), 29-63, arXiv:0803.2284.
  19. Strichartz R.S., Fractals in the large, Canad. J. Math. 50 (1998), 638-657.
  20. Strichartz R.S., Differential equations on fractals. A tutorial, Princeton University Press, Princeton, NJ, 2006.

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