Hostname: page-component-77c89778f8-n9wrp Total loading time: 0 Render date: 2024-07-22T12:52:12.088Z Has data issue: false hasContentIssue false
  • English
  • Français

Ring and module structures on dimension groups associated with a shift of finite type

Published online by Cambridge University Press:  01 June 2012

D. B. KILLOUGH  and
I. F. PUTNAM
D. B. KILLOUGH
Affiliation:
Department of Mathematics, Physics, and Engineering, Mount Royal University, Calgary, AB, T3E 6K6, Canada (email: bkillough@mtroyal.ca)
I. F. PUTNAM
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 3R4, Canada (email: ifputnam@uvic.ca)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study invariants for shifts of finite type obtained as the K-theory of various C*-algebras associated with them. These invariants have been studied intensively over the past thirty years since their introduction by Wolfgang Krieger. They may be given quite concrete descriptions as inductive limits of simplicially ordered free abelian groups. Shifts of finite type are special cases of Smale spaces and, in earlier work, the second author has shown that the hyperbolic structure of the dynamics in a Smale space induces natural ring and module structures on certain of these K-groups. Here, we restrict our attention to the special case of shifts of finite type and obtain explicit descriptions in terms of the inductive limits.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 4 , August 2012 , pp. 1370 - 1399
Copyright
Copyright © Cambridge University Press 2012

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

[1]Blackadar, B.. K-theory for Operator Algebras. Springer, Berlin, 1986.CrossRefGoogle Scholar
[2]Bowen, R.. On Axiom A Diffeomorphisms (CBMS Regional Conference Series in Mathematics, 35). American Mathematical Society, Providence, RI, 1978.Google Scholar
[3]Connes, A. and Higson, N.. Déformations, morphismes asymptotiques et K-théorie bivariante. C.R. Acad. Sci. Paris Sér. I 311 (1990), 101–106.Google Scholar
[4]Cuntz, J. and Krieger, W.. A class of C*-algebras and topological Markov chains. Invent. Math. 56 (1980), 251–268.CrossRefGoogle Scholar
[5]Handelman, D.. Positive matrices and dimension groups affiliated to C*-algebras and topological Markov chains. J. Operator Theory 6 (1981), 55–74.Google Scholar
[6]Handelman, D.. Free rank n+1 dense subgroups of ℝn and their endomorphisms. J. Funct. Anal. 46 (1982), 1–27.CrossRefGoogle Scholar
[7]Handelman, D.. Matrices of positive polynomials. Electronic J. Linear Algebra 19 (2009), 2–89.CrossRefGoogle Scholar
[8]Krieger, W.. On dimension functions and topological Markov chains. Invent. Math. 56 (1980), 239–250.CrossRefGoogle Scholar
[9]Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[10]Muhly, P. S., Renault, J. N. and Williams, D. P.. Equivalence and isomorphism for groupoid C *-algebas. J. Operator Theory 17 (1987), 3–22.Google Scholar
[11]Putnam, I. F.. C*-algebras from Smale spaces. Canad. J. Math. 48 (1996), 175–195.CrossRefGoogle Scholar
[12]Putnam, I. F.. Smale spaces and C*-algebras. Lecture Notes, Department of Mathematics and Statistics, University of Victoria, 2009.Google Scholar
[13]Putnam, I. F. and Spielberg, J.. The structure of C *-algebras associated with hyperbolic dynamical systems. J. Funct. Anal. 163 (1999), 279–299.CrossRefGoogle Scholar
[14]Rørdam, M., Larsen, F. and Laustsen, J.. An Introduction to K-Theory for C*-Algebras (London Mathematical Society Student Texts, 49). Cambridge University Press, Cambridge, 2000.CrossRefGoogle Scholar
[15]Ruelle, D.. Thermodynamic Formalism (Encyclopedia of Mathematics and its Applications, 5). Addison-Wesley, Reading, 1978.Google Scholar
[16]Ruelle, D.. Noncommutative algebras for hyperbolic diffeomorphisms. Invent. Math. 93 (1988), 1–13.CrossRefGoogle Scholar
[17]Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar