An Introduction to the C∗-Algebra of a One-Sided Shift Space

Conference paper
First Online:
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 58)

Abstract

This paper gives an introduction to the C ∗-algebra of a one-sided shift space. Focus will be given to the fundamental structure of the C ∗-algebra of a one-sided shift space, but some of the most important results about C ∗-algebras associated to shift spaces will also be presented.

Keywords

C∗-algebras of shift spaces C∗-algebras of subshifts 
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Notes

Acknowledgements

The author was supported by the NordForsk Research Network Operator Algebra and Dynamics (grant #11580) and the Research Council of Norway through project 191195/V30.

References

  1. 1.
    Arveson, W.: An invitation to C ∗-algebras. Springer-Verlag, New York (1976). Graduate Texts in Mathematics, No. 39Google Scholar
  2. 2.
    Bates, T., Carlsen, T.M., Eilers, S.: Dimension groups associated to β-expansions. Math. Scand. 100(2), 198–208 (2007)MathSciNetMATHGoogle Scholar
  3. 3.
    Blackadar, B.: K-theory for operator algebras, Mathematical Sciences Research Institute Publications, vol. 5, second edn. Cambridge University Press, Cambridge (1998)Google Scholar
  4. 4.
    Blackadar, B.: Operator algebras, Encyclopaedia of Mathematical Sciences, vol. 122. Springer-Verlag, Berlin (2006). Theory of C ∗-algebras and von Neumann algebras, Operator Algebras and Non-commutative Geometry, IIIGoogle Scholar
  5. 5.
    Boyle, M., Handelman, D.: Orbit equivalence, flow equivalence and ordered cohomology. Israel J. Math. 95, 169–210 (1996). DOI 10.1007/BF02761039. URL http://dx.doi.org/10.1007/BF02761039
  6. 6.
    Carlsen, T.M.: A faithful representation of the C ∗-algebra associated to a shift space. In preparationGoogle Scholar
  7. 7.
    Carlsen, T.M.: On C ∗-algebras associated with sofic shifts. J. Operator Theory 49(1), 203–212 (2003)MathSciNetMATHGoogle Scholar
  8. 8.
    Carlsen, T.M.: Operator algebraic applications in symbolic dynamics. Ph.D. thesis, University of Copenhagen (2004). URL http://www.math.ku.dk/noter/filer/phd04tmc.pdf
  9. 9.
    Carlsen, T.M.: C ∗-algebras associated to shift spaces (2008), arXiv:0808.0301v1Google Scholar
  10. 10.
    Carlsen, T.M.: Cuntz-Pimsner C ∗-algebras associated with subshifts. Internat. J. Math. 19(1), 47–70 (2008). DOI 10.1142/S0129167X0800456X. URL http://dx.doi.org/10.1142/S0129167X0800456X
  11. 11.
    Carlsen, T.M., Eilers, S.: Augmenting dimension group invariants for substitution dynamics. Ergodic Theory Dynam. Systems 24(4), 1015–1039 (2004). DOI 10.1017/ S0143385704000057. URL http://dx.doi.org/10.1017/S0143385704000057
  12. 12.
    Carlsen, T.M., Eilers, S.: Matsumoto K-groups associated to certain shift spaces. Doc. Math. 9, 639–671 (electronic) (2004)Google Scholar
  13. 13.
    Carlsen, T.M., Eilers, S.: Ordered K-groups associated to substitutional dynamics. J. Funct. Anal. 238(1), 99–117 (2006). DOI 10.1016/j.jfa.2005.12.028. URL http://dx.doi.org/10.1016/j.jfa.2005.12.028
  14. 14.
    Carlsen, T.M., Matsumoto, K.: Some remarks on the C ∗-algebras associated with subshifts. Math. Scand. 95(1), 145–160 (2004)MathSciNetMATHGoogle Scholar
  15. 15.
    Carlsen, T.M., Silvestrov, S.: C ∗-crossed products and shift spaces. Expo. Math. 25(4), 275–307 (2007). DOI 10.1016/j.exmath.2007.02.004. URL http://dx.doi.org/10.1016/j.exmath.2007.02.004 Google Scholar
  16. 16.
    Carlsen, T.M., Silvestrov, S.: On the K-theory of the C ∗-algebra associated with a one-sided shift space. Proc. Est. Acad. Sci. 59(4), 272–279 (2010). DOI 10.3176/proc.2010.4.04. URL http://dx.doi.org/10.3176/proc.2010.4.04
  17. 17.
    Cuntz, J., Krieger, W.: A class of C ∗-algebras and topological Markov chains. Invent. Math. 56(3), 251–268 (1980). DOI 10.1007/BF01390048. URL http://dx.doi.org/10.1007/BF01390048 Google Scholar
  18. 18.
    Franks, J.: Flow equivalence of subshifts of finite type. Ergodic Theory Dynam. Systems 4(1), 53–66 (1984). DOI 10.1017/S0143385700002261. URL http://dx.doi.org/10.1017/S0143385700002261
  19. 19.
    Ito, S., Takahashi, Y.: Markov subshifts and realization of β-expansions. J. Math. Soc. Japan 26, 33–55 (1974)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Katayama, Y., Matsumoto, K., Watatani, Y.: Simple C ∗-algebras arising from β-expansion of real numbers. Ergodic Theory Dynam. Systems 18(4), 937–962 (1998). DOI 10.1017/S0143385798108350. URL http://dx.doi.org/10.1017/S0143385798108350 Google Scholar
  21. 21.
    Katsura, T.: On C ∗-algebras associated with C ∗-correspondences. J. Funct. Anal. 217(2), 366–401 (2004). DOI 10.1016/j.jfa.2004.03.010. URL http://dx.doi.org/10.1016/j.jfa.2004.03.010 Google Scholar
  22. 22.
    Krieger, W., Matsumoto, K.: Shannon graphs, subshifts and lambda-graph systems. J. Math. Soc. Japan 54(4), 877–899 (2002). DOI 10.2969/jmsj/1191591995. URL http://dx.doi.org/10.2969/jmsj/1191591995 Google Scholar
  23. 23.
    Krieger, W., Matsumoto, K.: A lambda-graph system for the Dyck shift and its K-groups. Doc. Math. 8, 79–96 (electronic) (2003)Google Scholar
  24. 24.
    Lind, D., Marcus, B.: An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge (1995). DOI 10.1017/CBO9780511626302. URL http://dx.doi.org/10.1017/CBO9780511626302
  25. 25.
    Matsumoto, K.: On C ∗-algebras associated with subshifts. Internat. J. Math. 8(3), 357–374 (1997). DOI 10.1142/S0129167X97000172. URL http://dx.doi.org/10.1142/S0129167X97000172
  26. 26.
    Matsumoto, K.: Interpolated Cuntz algebras from β-expansions of real numbers. Sūrikaisekikenkyūsho Kōkyūroku (1024), 84–86 (1998). Profound development of operator algebras (Japanese) (Kyoto, 1997)Google Scholar
  27. 27.
    Matsumoto, K.: K-theory for C ∗-algebras associated with subshifts. Math. Scand. 82(2), 237–255 (1998)MathSciNetGoogle Scholar
  28. 28.
    Matsumoto, K.: Dimension groups for subshifts and simplicity of the associated C ∗-algebras. J. Math. Soc. Japan 51(3), 679–698 (1999). DOI 10.2969/jmsj/05130679. URL http://dx.doi.org/10.2969/jmsj/05130679 Google Scholar
  29. 29.
    Matsumoto, K.: Presentations of subshifts and their topological conjugacy invariants. Doc. Math. 4, 285–340 (electronic) (1999)Google Scholar
  30. 30.
    Matsumoto, K.: Relations among generators of C ∗-algebras associated with subshifts. Internat. J. Math. 10(3), 385–405 (1999). DOI 10.1142/S0129167X99000148. URL http://dx.doi.org/10.1142/S0129167X99000148
  31. 31.
    Matsumoto, K.: A simple C ∗-algebra arising from a certain subshift. J. Operator Theory 42(2), 351–370 (1999)MathSciNetMATHGoogle Scholar
  32. 32.
    Matsumoto, K.: On automorphisms of C ∗-algebras associated with subshifts. J. Operator Theory 44(1), 91–112 (2000)MathSciNetMATHGoogle Scholar
  33. 33.
    Matsumoto, K.: Stabilized C ∗-algebras constructed from symbolic dynamical systems. Ergodic Theory Dynam. Systems 20(3), 821–841 (2000). DOI 10.1017/S0143385700000444. URL http://dx.doi.org/10.1017/S0143385700000444
  34. 34.
    Matsumoto, K.: Bowen-Franks groups as an invariant for flow equivalence of subshifts. Ergodic Theory Dynam. Systems 21(6), 1831–1842 (2001). DOI 10.1017/ S0143385701001870. URL http://dx.doi.org/10.1017/S0143385701001870
  35. 35.
    Matsumoto, K.: Bowen-Franks groups for subshifts and Ext-groups for C ∗-algebras. K-Theory 23(1), 67–104 (2001). DOI 10.1023/A:1017568715542. URL http://dx.doi.org/10.1023/A:1017568715542
  36. 36.
    Matsumoto, K.: Strong shift equivalence of symbolic dynamical systems and Morita equivalence of C ∗-algebras. Ergodic Theory Dynam. Systems 24(1), 199–215 (2004). DOI 10.1017/S014338570300018X. URL http://dx.doi.org/10.1017/S014338570300018X
  37. 37.
    Matsumoto, K., Watatani, Y., Yoshida, M.: KMS states for gauge actions on C ∗-algebras associated with subshifts. Math. Z. 228(3), 489–509 (1998). DOI 10.1007/PL00004627. URL http://dx.doi.org/10.1007/PL00004627 Google Scholar
  38. 38.
    Murphy, G.J.: C ∗-algebras and operator theory. Academic Press Inc., Boston, MA (1990)MATHGoogle Scholar
  39. 39.
    Parry, B., Sullivan, D.: A topological invariant of flows on 1-dimensional spaces. Topology 14(4), 297–299 (1975)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Raeburn, I., Williams, D.P.: Morita equivalence and continuous-trace C ∗-algebras, Mathematical Surveys and Monographs, vol. 60. American Mathematical Society, Providence, RI (1998)CrossRefGoogle Scholar
  41. 41.
    Rørdam, M., Larsen, F., Laustsen, N.: An introduction to K-theory for C ∗-algebras, London Mathematical Society Student Texts, vol. 49. Cambridge University Press, Cambridge (2000)Google Scholar
  42. 42.
    Wegge-Olsen, N.E.: K-theory and C ∗-algebras. A friendly approach. Oxford Science Publications. The Clarendon Press Oxford University Press, New York (1993)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorwegian University of Science and Technology (NTNU)TrondheimNorway

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