Hostname: page-component-6bf8c574d5-956mj Total loading time: 0 Render date: 2025-02-21T04:34:48.375Z Has data issue: false hasContentIssue false
  • English
  • Français

The structure of the C*-algebra of a locally injective surjection

Published online by Cambridge University Press:  10 June 2011

TOKE MEIER CARLSEN  and
KLAUS THOMSEN
TOKE MEIER CARLSEN
Affiliation:
Department of Mathematics, Norwegian University of Science and Technology, NO-7034 Trondheim, Norway (email: tokemeie@math.ntnu.no)
KLAUS THOMSEN
Affiliation:
Institut for matematiske fag, Ny Munkegade, DK-8000 Aarhus C, Denmark (email: matkt@imf.au.dk)
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.

In this paper we investigate the ideal structure of the C*-algebra of a locally injective surjection introduced by the second-named author. Our main result is that a simple quotient of the C*-algebra of a locally injective surjection on a compact metric space of finite covering dimension is either a full matrix algebra, a crossed product of a minimal homeomorphism of a compact metric space of finite covering dimension, or it is purely infinite and hence covered by the classification result of Kirchberg and Phillips. It follows in particular that if the C*-algebra of a locally injective surjection on a compact metric space of finite covering dimension is simple, then it is automatically purely infinite, unless the map in question is a homeomorphism. A corollary of this result is that if the C*-algebra of a one-sided subshift is simple, then it is also purely infinite.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 4 , August 2012 , pp. 1226 - 1248
Copyright
Copyright © Cambridge University Press 2011

References

[An]Anantharaman-Delaroche, C.. Purely infinite C *-algebras arising from dynamical systems. Bull. Soc. Math. France 125 (1997), 199–225.Google Scholar
[BKR]Blackadar, B., Kumjian, A. and Rørdam, M.. Approximately central matrix units and the structure of noncommative tori. K-theory 6 (1992), 267–284.Google Scholar
[BoKR]Boyd, S., Keswani, N. and Raeburn, I.. Faithful representations of crossed products by endomorphisms. Proc. Amer. Math. Soc. 118 (1993), 427–436.Google Scholar
[BL]Boyle, M. and Lind, D.. Expansive subdynamics. Trans. Amer. Math. Soc. 349 (1997), 55–102.CrossRefGoogle Scholar
[BS]Brin, M. and Stuck, G.. Introduction to Dynamical Systems. Cambridge University Press, Cambridge, 2002.CrossRefGoogle Scholar
[Br]Brown, L.. Stable isomorphism of hereditary subalgebras of C *-algebras. Pacific J. Math. 71 (1977), 335–348.CrossRefGoogle Scholar
[Ca]Carlsen, T. M.. Cuntz–Pimsner C *-algebras associated with subshifts. Internat. J. Math. 19 (2008), 47–70.CrossRefGoogle Scholar
[De1]Deaconu, V.. Groupoids associated with endomorphisms. Trans. Amer. Math. Soc. 347 (1995), 1779–1786.CrossRefGoogle Scholar
[De2]Deaconu, V.. Generalized solenoids and C *-algebras. Pacific J. Math. 190 (1999), 247–260.Google Scholar
[DS]Deaconu, V. and Schultz, F.. C *-algebras associated with interval maps. Trans. Amer. Math. Soc. 359 (2007), 1889–1924.Google Scholar
[En]Engelking, R.. Dimension Theory. North-Holland, Amsterdam, 1978.Google Scholar
[Ka]Katsura, T.. A class of C *-algebras generalizing both graph algebras and homeomorphism algebras. III Ideal structures. Ergod. Th. & Dynam. Sys. 26 (2006), 1805–1854.Google Scholar
[Ma]Matsumoto, K.. C *-algebras associated with presentations of subshifts. Doc. Math. 7 (2002), 1–30.Google Scholar
[Pe]Pedersen, G. K.. C *-algebras and their Automorphism Groups. Academic Press, New York, 1979.Google Scholar
[RW]Raeburn, I. and Williams, D. P.. Morita Equivalence and Continuous-Trace C *-algebras. American Mathematical Society, Providence, RI, 1998.CrossRefGoogle Scholar
[Re]Renault, J.. A Groupoid Approach to C *-algebras (Lecture Notes in Mathematics, 793). Springer, Berlin, 1980.Google Scholar
[St]Stacey, P. J.. Crossed products of C *-algebras by endomorphisms. J. Aust. Math. Soc. 54 (1993), 204–212.CrossRefGoogle Scholar
[Th1]Thomsen, K.. Semi-étale groupoids and applications. Ann. Inst. Fourier 60 (2010), 759–800.CrossRefGoogle Scholar
[Th2]Thomsen, K.. On the C *-algebra of a locally injective surjection and its KMS states. Comm. Math. Phys. 302 (2011), 403–423.Google Scholar
[Th3]Thomsen, K.. Pure infiniteness of the crossed product of an AH-algebra by an endomorphism. arXiv:1010.0960.v2 (2010), submitted for publication, 14 pp.Google Scholar