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C*-algebras associated with presentations of subshifts ii. ideal structure and lambda-graph subsystems

Part of: Selfadjoint operator algebras Topological dynamics

Published online by Cambridge University Press:  09 April 2009

Kengo Matsumoto
Kengo Matsumoto
Affiliation:
Department of Mathematical SciencesYokohama City UniversitySeto 22-2, Kanazawa-ku Yokohama 236-0027Japan e-mail: kengo@yokohama-cu.ac.jp
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Abstract

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A λ-graph system is a labeled Bratteli diagram with shift transformation. It is a generalization of finite labeled graphs and presents a subshift. In Doc. Math. 7 (2002) 1–30, the author constructed a C*-algebra O£ associated with a λ-graph system £ from a graph theoretic view-point. If a λ-graph system comes from a finite labeled graph, the algebra becomes a Cuntz-Krieger algebra. In this paper, we prove that there is a bijective correspondence between the lattice of all saturated hereditary subsets of £ and the lattice of all ideals of the algebra O£, under a certain condition on £ called (II). As a result, the class of the C*-algebras associated with λ-graph systems under condition (II) is closed under quotients by its ideals.

MSC classification

Secondary: 46L35: Classifications of $C^*$-algebras 46L05: General theory of $C^*$-algebras 37B10: Symbolic dynamics
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 81 , Issue 3 , December 2006 , pp. 369 - 385
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Aho, A. V. and Ullman, J., ‘The theory of language’, Math. Systems Theory 2 (1968), 97–125.CrossRefGoogle Scholar
[2]Bates, T., Pask, D., Raeburn, I. and Szymański, W., ‘The C*-algebras of row-finite graphs’, New York J. Math. 6 (2000), 307–324.Google Scholar
[3]Blackadar, B., ‘Semi-projectivity in simple C*-algebras’, Adv. Stud. Pure Math. 38 (2004), 1–17.Google Scholar
[4]Brown, L. G., Green, P. and Rieffel, M. A., ‘Stable isomorphism and strong Morita equivalence of C*-algebras’, Pacific. J. Math. 71 (1977), 349–363.CrossRefGoogle Scholar
[5]Carlsen, T. M. and Matsumoto, K., ‘Some remarks on the C*-algebras associated with subshifts’, Math. Scand. 95 (2004), 145–160.CrossRefGoogle Scholar
[6]Cuntz, J., ‘A class of C*-algebras and topological Markov chains II: reducible chains and the Extfunctor for C*-algebras’, Invent. Math. 63 (1980), 25–40.CrossRefGoogle Scholar
[7]Cuntz, J. and Krieger, W., ‘A class of C*-algebras and topological Markov chains’, Invent. Math. 56 (1980), 251–268.CrossRefGoogle Scholar
[8]Deaconu, V., ‘Groupoids associated with endomorphisms’, Trans. Amer. Math. Soc. 347 (1995), 1779–1786.CrossRefGoogle Scholar
[9]Deaconu, V., ‘Generalized solenoids and C*-algebras’, Pacific. J. Math. 190 (1999), 247–260.CrossRefGoogle Scholar
[10]Excel, R. and Laca, M., ‘Cuntz-Krieger algebras for infinite matrices’, J. Reine. Angew. Math. 512 (1999), 119–172.CrossRefGoogle Scholar
[11]Fiebig, D. and Fiebig, U.-R., ‘Covers for coded systems’, Contemp. Math. 135 (1992), 139–180.CrossRefGoogle Scholar
[12]Fowler, N., Laca, M. and Raeburn, I., ‘The C*-algebras of infinite graphs’, Proc. Amer. Math. Soc. 8 (2000), 2319–2327.Google Scholar
[13]an Huef, A. and Raeburn, I., ‘The ideal structure of Cuntz-Krieger algebras’, Ergodic Theory Dynam. Systems 17 (1997), 611–624.CrossRefGoogle Scholar
[14]Kajiwara, T., Pinzari, C. and Watatani, Y., ‘Ideal structure and simplicity of the C*-algebras generated by Hilbert modules’, J. Funct. Anal. 159 (1998), 295–322.CrossRefGoogle Scholar
[15]Katsura, T., ‘A class of C*-algebras generalizing both graph algebras and homeomorphism C*-algebras I, fundamental results’, Trans. Amer. Math. Soc. 356 (2004), 4287–4322.CrossRefGoogle Scholar
[16]Krieger, W. and Matsumoto, K., ‘A lambda-graph system for the dyck shift and its K-groups’, Doc. Math. 8 (2003), 79–96.CrossRefGoogle Scholar
[17]Kumjian, A., Pask, D. and Raeburn, I., ‘Cuntz-Krieger algebras of directed graphs’, Pacific J. Math. 184 (1998), 161–174.CrossRefGoogle Scholar
[18]Kumjian, A., Pask, D., Raeburn, I. and Renault, J., ‘Graphs, groupoids and Cuntz-Krieger algebras’, J. Funct. Anal. 144 (1997), 505–541.CrossRefGoogle Scholar
[19]Lind, D. and Marcus, B., An introduction to symbolic dynamics and coding (Cambridge University Press, Cambridge, 1995).CrossRefGoogle Scholar
[20]Matsumoto, K., ‘On the simple C*-algebras arising from Dyck systems’, J. Operator Theory to appear.Google Scholar
[21]Matsumoto, K., ‘On C*-algebras associated with subshifts’, Internat. J. Math. 8 (1997), 357–374.CrossRefGoogle Scholar
[22]Matsumoto, K., ‘K-theory for C*-algebras associated with subshifts’, Math. Scand. 82 (1998), 237–255.CrossRefGoogle Scholar
[23]Matsumoto, K., ‘Presentations of subshifts and their topological conjugacy invariants’, Doc. Math. 4 (1999), 285–340.CrossRefGoogle Scholar
[24]Matsumoto, K., ‘C*-algebras associated with presentations of subshifts’, Doc. Math. 7 (2002), 1–30.CrossRefGoogle Scholar
[25]Matsumoto, K., ‘Symbolic dynamics and C*-algebras’, in: Selected papers on analysis and differential equations, Amer. Math. Soc. Transl. Ser. 211 (Amer. Math. Soc., Providence, RI, 2003) pp. 47–70.Google Scholar
[26]Matsumoto, K., ‘A simple purely infinite C*-algebra associated with a lambda-graph system of the Motzkin shift’, Math. Z. 248 (2004), 369–394.CrossRefGoogle Scholar
[27]Matsumoto, K., ‘Construction and pure infiniteness of the C*-algebras associated with λ-graph systems’, Math. Scand. 97 (2005), 73–88.CrossRefGoogle Scholar
[28]Matsumoto, K., ‘Factor maps of symbolic dynamics and inclusions of C*-algebras’, Internat. J. Math. 14 (2005), 313–339.Google Scholar
[29]Matsumoto, K., ‘K-theoretic invariants and conformal measures of the Dyck shifts’, Internat. J. Math. 16 (2005), 213–248.CrossRefGoogle Scholar
[30]Muhly, P. S. and Solel, B., ‘On the simplicity of some Cuntz-Pimsner algebras’, Math. Scand. 83 (1998), 53–73.CrossRefGoogle Scholar
[31]Renault, J. N., A groupoid approach to C*-algebras, Lecture Notes in Math. 793 (Springer, Berlin, 1980).CrossRefGoogle Scholar
[32]Rieffel, M. A., ‘Induced representations of C*-algebras’, Adv. Math. 13 (1974), 176–257.CrossRefGoogle Scholar
[33]Rørdam, M., ‘Classification of Cuntz-Krieger algebras’, K-Theory 9 (1995), 31–58.CrossRefGoogle Scholar
[34]Tomforde, M., ‘A unified approach to Exel-Laca algebras and C*-algebras associated to graphs’, J. Operator Theory 50 (2003), 345–368.Google Scholar
[35]Watatani, Y., ‘Graph theory for C*-algebras’, in: Operator Algebras and their Applications (ed. Kadison, R. V.), Proc. Sympos. Pure Math. 38, part 1 (Amer. Math. Soc., Providence, RI, 1982) pp. 195–197.CrossRefGoogle Scholar