Skip to main content Accessibility help
×
Home

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

  • TSUYOSHI KAJIWARA (a1) and YASUO WATATANI (a2)

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.

Copyright

References

Hide All
[1]Blackadar, B.. Operator Algebras (Encyclopedia of Mathematical Sciences, 122). Springer, Berlin, 2006.
[2]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. 2738.
[3]Combes, F. and Zettl, H.. Order structures, traces and weights on Morita equivalent ${C}^{\ast } $-algebras. Math. Ann. 265 (1983), 6781.
[4]Cuntz, J. and Krieger, W.. A class of ${C}^{\ast } $-algebras and topological Markov chains. Invent. Math. 56 (1980), 251268.
[5]Exel, R. and Laca, M.. Partial dynamical systems and the KMS condition. Comm. Math. Phys. 232 (2003), 223277.
[6]Falconer, K. J.. Fractal Geometry. Wiley, Chichester, 1997.
[7]Frank, M. and Larson, D.. Frames in Hilbert ${C}^{\ast } $-modules and ${C}^{\ast } $-algebras. J. Operator Theory 48 (2002), 273314.
[8]Hutchinson, J. E.. Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713747.
[9]Ionescu, M. and Watatani, Y.. ${C}^{\ast } $-algebras associated with Mauldin–Williams graphs. Canad. Math. Bull. 51 (2008), 545560.
[10]Izumi, M., Kajiwara, T. and Watatani, Y.. KMS states and branched points. Ergod. Th. & Dynam. Sys. 27 (2007), 18871918.
[11]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. 7391.
[12]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), 295322.
[13]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), 149.
[14]Kajiwara, T. and Watatani, Y.. C*-algebras associated with self-similar sets. J. Operator Theory 56 (2006), 225247.
[15]Kajiwara, T. and Watatani, Y.. KMS states on finite-graph C*-algebras. Kyushu J. Math. 67 (2013), 83104.
[16]Kigami, J.. Analysis on Fractals. Cambridge University Press, Cambridge, 2001.
[17]Kumjian, A. and Renalut, J.. KMS states on ${C}^{\ast } $-algebras associated to expansive maps. Proc. Amer. Math. Soc. 134 (2006), 20672078.
[18]Laca, M. and Neshveyev, S.. KMS states of quasi-free dynamics on Pimsner algebras. J. Funct. Anal. 211 (2004), 457482.
[19]Matsumoto, K.. K-theory for ${C}^{\ast } $-algebras associated with subshifts. Math. Scand. 82 (1998), 237255.
[20]Nawata, N.. Fundamental group of simple C*-algebras with unique trace III. Canad. J. Math. 64 (2012), 573587.
[21]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. 189212.

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed