Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T18:48:56.640Z Has data issue: false hasContentIssue false
  • English
  • Français

C*-Algebras Associated with Mauldin–Williams Graphs

Published online by Cambridge University Press:  20 November 2018

Marius Ionescu  and
Yasuo Watatani
Marius Ionescu
Affiliation:
Department of Mathematics, Dartmouth College, Hanover, NH, U.S.A.andDepartment of Mathematics, Cornell University, Ithica, NY, U.S.A.
Yasuo Watatani
Affiliation:
Department of Mathematical Sciences, Kyushu University, Hakozaki, Fukuoka, 812-8581, Japan
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A Mauldin–Williams graph $M$ is a generalization of an iterated function system by a directed graph. Its invariant set $K$ plays the role of the self-similar set. We associate a ${{C}^{*}}$-algebra ${{O}_{M}}\left( K \right)$ with a Mauldin–Williams graph $M$ and the invariant set $K$, laying emphasis on the singular points. We assume that the underlying graph $G$ has no sinks and no sources. If $M$ satisfies the open set condition in $K$, and $G$ is irreducible and is not a cyclic permutation, then the associated ${{C}^{*}}$-algebra ${{O}_{M}}\left( K \right)$ is simple and purely infinite. We calculate the $K$-groups for some examples including the inflation rule of the Penrose tilings.

Keywords

46L3546L0846L8037B10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 51 , Issue 4 , 01 December 2008 , pp. 545 - 560
Copyright
Copyright © Canadian Mathematical Society 2008

References

[1] Anderson, J. and Putnam, I., Topological invariants for substitution tilings and their associated C*-algebras. Ergodic Theory Dynam. Systems 18(1998), 509–537.Google Scholar
[2] Barnsley, M. F., Fractals Everywhere. Second edition. Academic Press Professional, Boston, MA, 1993.Google Scholar
[3] Bartholdi, L., Grigorchuk, R., and Nekrashevych, V., From fractal groups to fractal sets. In: Trends in Mathematics: Fractals in Graz 2001, Birkhäuser, Basel, 2003, pp. 25–118.Google Scholar
[4] Bratteli, O. and Jorgensen, P. E. T., Iterated function systems and permutation representations of the Cuntz algebra. Mem. Amer. Math. Soc. 139(1999), no. 663.Google Scholar
[5] Brenken, B., C*-algebras associated with topological relations. J. Ramanujan Math. Soc. 19(2004), no. 1, 35–55.Google Scholar
[6] Connes, A., Non-commutative Geometry, Academic Press, 1994.Google Scholar
[7] Cuntz, J., Simple C*-algebras generated by isometries. Comm. Math. Phys. 57(1977), no. 2, 173–185.Google Scholar
[8] Cuntz, J. and Krieger, W., A class of C*-algebras and topological Markov chains. Invent. Math. 56(1980), no. 3, 251–268.Google Scholar
[9] Deaconu, V. and Shultz, F., C*-algebras associated with interval maps. Trans. Amer. Math. Soc. 359(2007), no. 4, 1889–1924(electronic).Google Scholar
[10] Edgar, G. A., Measure, topology, and fractal geometry. Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1990.Google Scholar
[11] Edgar, G. A. and Mauldin, R. D., Multifractal decompositions of digraph recursive fractals. Proc. London Math. Soc. 65(1992), no. 3, 604–628.Google Scholar
[12] Fowler, N. J., Muhly, P. S. and Raeburn, I., Representations of Cuntz–Pimsner Algebras. Indiana Univ. Math. J. 52(2003), no. 3, 569–605.Google Scholar
[13] Hutchinson, J. E., Fractals and self-similarity. Indiana Univ. Math. J. 30(1981), no. 5, 713–747.Google Scholar
[14] Ionescu, M., Operator Algebras and Mauldin–Williams graphs. Rocky Mountain J. Math. 37(2007), no. 3, 829–849.Google Scholar
[15] Ionescu, M.,Maudin-Williams graphs, Morita equivalence and isomorphisms. Proc. Amer. Math. Soc. 134(2006), no. 4, 1087–1097.Google Scholar
[16] Kajiwara, T. and Watatani, Y., ‘C*-algebras associated with self-similar sets. J. Operator Theory 56(2006), no. 2, 225–247.Google Scholar
[17] Kajiwara, T. and Watatani, Y., C*-algebras associated with complex dynamical systems. Indiana Univ. Math. J. 54(2005), no. 3, 755–778.Google Scholar
[18] Katsura, T., A class of C*-algebras generalizing both graph algebras and homeomorphism C*-algebras. I. Fundamental results. Trans. Amer. Math. Soc. 356(2004), no. 11, 4287–4322.Google Scholar
[19] Kigami, J., Analysis on Fractals, Cambridge Tracts in Mathematics 143. Cambridge University Press, Cambridge, 2001.Google Scholar
[20] Kirchberg, E., The classification of purely infinite C*-algebras using Kasparov's theory, preprint.Google Scholar
[21] Kumjian, A., Pask, D. and Raeburn, I., Cuntz–Krieger algebras of directed graphs. Pacific J. Math. 184(1998), no. 1, 161–174.Google Scholar
[22] Lance, E. C., Hilbert C*-modules. A toolkit for operator algebraists. London Mathematical Society Lecture Note Series 210, Cambridge University Press, Cambridge, 1995.Google Scholar
[23] Mauldin, R. D. and Williams, S. C., Hausdorff dimension in graph directed constructions. Trans. Amer. Math. Soc. 309(1988), no. 2, 811–829.Google Scholar
[24] Mingo, J., ‘C*-algebras associated with one-dimensional almost periodic tilings. Comm. Math. Phys. 183(1997), no. 2, 307–337.Google Scholar
[25] Muhly, P. S. and Solel, B., Tensor algebras over C*-correspondences: representations, dilations, and C*-envelopes. J. Funct. Anal. 158(1998), no. 2, 389–457.Google Scholar
[26] Muhly, P. S. and Solel, B., On the Morita equivalence of tensor algebras. Proc. London Math. Soc. 81(2000), no. 1, 113–168.Google Scholar
[27] Muhly, P. and Tomforde, M., Topological quivers. Internat. J. Math. 16(2005), no. 7, 693–755.Google Scholar
[28] Phillips, N. C., A classification theorem for nuclear purely infinite simple C*-algebras. Doc. Math. 5(2000), 49–114 (electronic).Google Scholar
[29] Pimsner, M. V., A class of C*-algebras generalizing both Cuntz–Krieger algebras and crossed products by ℤ’. In: Free Probability Theory. 189–212, Fields Inst. Commun. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 189–212.Google Scholar
[30] Pinzari, C., Watatani, Y., and Yonetani, K., KMS states, entropy and the variational principle in full C*-dynamical systems. Comm. Math. Phys. 213(2000), no. 2, 331–379.Google Scholar