Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T04:48:40.009Z Has data issue: false hasContentIssue false
  • English
  • Français

Traces on cores of ${C}^{\ast } $-algebras associated with self-similar maps

Published online by Cambridge University Press:  05 July 2013

TSUYOSHI KAJIWARA  and
YASUO WATATANI
TSUYOSHI KAJIWARA
Affiliation:
Department of Environmental and Mathematical Sciences, Okayama University, Tsushima, Okayama 700-8530, Japan email [email protected]
YASUO WATATANI
Affiliation:
Department of Mathematical Sciences, Kyushu University, Motooka, Fukuoka 819-0395, Japan email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We completely classify the extreme tracial states on the cores of the ${C}^{\ast } $-algebras associated with self-similar maps on compact metric spaces. We present a complete list of them. The extreme tracial states are the union of the discrete type tracial states given by measures supported on the finite orbits of the branch points and a continuous type tracial state given by the Hutchinson measure on the original self-similar set.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 6 , December 2014 , pp. 1964 - 1989
Copyright
© Cambridge University Press, 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Blackadar, B.. Operator Algebras (Encyclopedia of Mathematical Sciences, 122). Springer, Berlin, 2006.Google Scholar
Castro, G.. ${C}^{\ast } $-algebras associated with iterated function systems. Operator Structures and Dynamical Systems (Contemporary Mathematics, 503). American Mathematical Society, Providence, RI, 2009, pp. 27–38.Google Scholar
Combes, F. and Zettl, H.. Order structures, traces and weights on Morita equivalent ${C}^{\ast } $-algebras. Math. Ann. 265 (1983), 67–81.Google Scholar
Cuntz, J. and Krieger, W.. A class of ${C}^{\ast } $-algebras and topological Markov chains. Invent. Math. 56 (1980), 251–268.CrossRefGoogle Scholar
Exel, R. and Laca, M.. Partial dynamical systems and the KMS condition. Comm. Math. Phys. 232 (2003), 223–277.Google Scholar
Falconer, K. J.. Fractal Geometry. Wiley, Chichester, 1997.Google Scholar
Frank, M. and Larson, D.. Frames in Hilbert ${C}^{\ast } $-modules and ${C}^{\ast } $-algebras. J. Operator Theory 48 (2002), 273–314.Google Scholar
Hutchinson, J. E.. Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713–747.CrossRefGoogle Scholar
Ionescu, M. and Watatani, Y.. ${C}^{\ast } $-algebras associated with Mauldin–Williams graphs. Canad. Math. Bull. 51 (2008), 545–560.Google Scholar
Izumi, M., Kajiwara, T. and Watatani, Y.. KMS states and branched points. Ergod. Th. & Dynam. Sys. 27 (2007), 1887–1918.Google Scholar
Kajiwara, T.. Countable bases for Hilbert ${C}^{\ast } $-modules and classification of KMS states. Operator Structures and Dynamical Systems (Contemporary Mathematics, 503). American Mathematical Society, Providence, RI, 2009, pp. 73–91.Google Scholar
Kajiwara, T., Pinzari, C. and Watatani, Y.. Ideal structure and simplicity of the ${C}^{\ast } $-algebras generated by Hilbert bimodules. J. Funct. Anal. 159 (1998), 295–322.CrossRefGoogle Scholar
Kajiwara, T., Pinzari, C. and Watatani, Y.. Jones index theory for Hilbert ${C}^{\ast } $-bimodules and its equivalence with conjugation theory. J. Funct. Anal. 215 (2004), 1–49.Google Scholar
Kajiwara, T. and Watatani, Y.. C*-algebras associated with self-similar sets. J. Operator Theory 56 (2006), 225–247.Google Scholar
Kajiwara, T. and Watatani, Y.. KMS states on finite-graph C*-algebras. Kyushu J. Math. 67 (2013), 83–104.CrossRefGoogle Scholar
Kigami, J.. Analysis on Fractals. Cambridge University Press, Cambridge, 2001.Google Scholar
Kumjian, A. and Renalut, J.. KMS states on ${C}^{\ast } $-algebras associated to expansive maps. Proc. Amer. Math. Soc. 134 (2006), 2067–2078.Google Scholar
Laca, M. and Neshveyev, S.. KMS states of quasi-free dynamics on Pimsner algebras. J. Funct. Anal. 211 (2004), 457–482.Google Scholar
Matsumoto, K.. K-theory for ${C}^{\ast } $-algebras associated with subshifts. Math. Scand. 82 (1998), 237–255.Google Scholar
Nawata, N.. Fundamental group of simple C*-algebras with unique trace III. Canad. J. Math. 64 (2012), 573–587.Google Scholar
Pimsner, M.. A class of ${C}^{\ast } $-algebras generating both Cuntz–Krieger algebras and crossed product by $ \mathbb{Z} $. Free Probability Theory. American Mathematical Society, Providence, RI, 1997, pp. 189–212.Google Scholar