Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-26T01:05:17.771Z Has data issue: false hasContentIssue false
  • English
  • Français

A Class of Limit Algebras Associated with Directed Graphs

Part of: Selfadjoint operator algebras Linear spaces and algebras of operators

Published online by Cambridge University Press:  09 April 2009

David W. Kribs  and
Baruch Solel
David W. Kribs
Affiliation:
Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario NIG 2W1, Canadadkribs@uoguelph.ca
Baruch Solel
Affiliation:
Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israelmabaruch@techunix.technion.ac.il
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.

Every directed graph defines a Hilbert space and a family of weighted shifts that act on the space. We identify a natural notion of periodicity for such shifts and study their C* -algebras. We prove the algebras generated by all shifts of a fixed period are of Cuntz-Krieger and Toeplitz-Cuntz-Krieger type. The limit C* -algebras determined by an increasing sequence of positive integers, each dividing the next, are proved to be isomorphic to Cuntz-Pimsner algebras and the linking maps are shown to arise as factor maps. We derive a characterization of simplicity and compute the K-groups for these algebras. We prove a classification theorem for the class of algebras generated by simple loop graphs.

MSC classification

Secondary: 46L05: General theory of $C^*$-algebras 47L40: Limit algebras, subalgebras of $C^*$-algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 82 , Issue 3 , June 2007 , pp. 345 - 368
Copyright
Copyright © Australian Mathematical Society 2007

References

[1] Archbold, R. J., ‘An averaging process for C* -algebras related to weighted shifts’, Proc. London Math. Soc. (3) 35 (1977), 541–554.CrossRefGoogle Scholar
[2] Brenken, B., ‘C* -algebras associated with topological relations’, J. Ramanujan Math. Soc. 19 (2004), 35–55.Google Scholar
[3] Bunce, J. and Deddens, J., ‘C* -algebras generated by weighted shifts’, Indiana Univ. Math. J. 23 (1973), 257–271.CrossRefGoogle Scholar
[4] Bunce, J. and Deddens, J., ‘A family of simple C* -algebras related to weighted shift operators’, J. Funct. Anal. 19 (1975), 13–24.CrossRefGoogle Scholar
[5] Dadarlat, M. and Gong, G., ‘A classification result for approximately homogeneous C* -algebras of real rank zero’, Geom. Funct. Anal. 7 (1997), 646–711.CrossRefGoogle Scholar
[6] Davidson, K., C* -algebras by example, Fields Institute Monographs 6 (Amer. Math. Soc, Providence, RI, 1996).Google Scholar
[7] Elliott, G. A., Gong, G., Lin, H. and Pasnicu, C., ‘Abelian C* -subalgebrasof C* -algebras of real rank zero and inductive limit C* -algebras’, Duke Math. J. 85 (1996). 511–554.CrossRefGoogle Scholar
[8] Evans, D., ‘Gauge actions on OA’ J. Operator Theory 7 (1982), 79–100.Google Scholar
[9] Fowler, N., Muhly, P. and Raeburn, I., ‘Representations of Cuntz-Pimsner algebras’, Indiana Univ. Math. J. 52 (2003), 569–605.CrossRefGoogle Scholar
[10] Fowler, N. and Raeburn, I., ‘The Toeplitz algebra of a Hilbert bimodule’, Indiana Univ. Math. J. 48 (1999), 155–181.CrossRefGoogle Scholar
[11] 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
[12] Katsura, T., ‘A class of C* -algebras generalizing both graph algebras and homeomorphism C* -algebras. II. Examples’, Internat. J. Math. 17 (2006), 791–833.CrossRefGoogle Scholar
[13] Katsura, T., ‘A class of C* -algebras generalizing both graph algebras and homeomorphism C* -algebras. III. Ideal structures’, Ergodic Theory Dyn. Sys. 26 (2006), 1805–1854.CrossRefGoogle Scholar
[14] Kribs, D. W., ‘Inductive limit algebras from periodic weighted shifts on fock space’, New York J. Math. 8 (2002), 145–159.Google Scholar
[15] Kribs, D. W., ‘On bilateral weighted shifts in noncommutative multivariable operator theory’, Indiana Univ. Math. J. 52 (2003), 1595–1614.CrossRefGoogle Scholar
[16] Kribs, D. W. and Solel, B., ‘A class of limit algebras associated with directed graphs’, Preprint (math.OA/0411379).Google Scholar
[17] Muhly, P. and Solel, B., ‘Tensor algebras over C* -correspondences: representations, dilations, and C* -envelopes’, J. Fund. Anal. 158 (1998), 389–457.CrossRefGoogle Scholar
[18] Muhly, P. and Solel, B., ‘Tensor algebras, induced representations, and the Wold decomposition’, Canad. J. Math. 51(1999), 850–880.CrossRefGoogle Scholar
[19] Muhly, P. and Solel, B., ‘On the Morita equivalence of tensor algebras’, Proc. London Math. Soc. (3) 81 (2000), 113–168.CrossRefGoogle Scholar
[20] Muhly, P. and Tomforde, M., ‘Topological quivers’, Internat. J. Math. 16 (2005), 693–755.CrossRefGoogle Scholar
[21] O'uchi, M., ‘C* -bundles associated with generalized Bratteli diagrams’, Internat. J. Math. 9 (1998), 95–105.CrossRefGoogle Scholar
[22] Pasnicu, C., ‘Automorphisms of inductive limit C* -algebras’, Math. Scand. 74 (1994), 263–270.CrossRefGoogle Scholar
[23] Pimsner, M., ‘A class of C* -algebras generalizing both Cuntz-Krieger algebras and crossed products by ’, in: Free probability theory (Waterloo, ON, 1995). Fields Inst. Commun. 12 (Amer. Math. Soc., Providence, RI, 1997) pp. 189–212.Google Scholar
[24] Power, S. C., ‘Non-self-adjoint operator algebras and inverse systems of simplicial complexes’, J. Reine Angew. Math. 421 (1991), 43–61.Google Scholar
[25] Raeburn, I., Graph algebras, Conference Board of the Mathematical Sciences (Amer. Math. Soc, Providence, RI, 2005).Google Scholar
[26] Rordam, M., ‘Classification of nuclear, simple C* -algebras’, in: Classification of nuclear C* -algebras. Entropy in operator algebras. Encyclopedia Math. Sci. 126 (Springer, Berlin, 2002) pp. 1–145.CrossRefGoogle Scholar
[27] Solel, B., ‘Limit algebras associated with an automorphism’, Math. Scand. 95 (2004), 101–123.CrossRefGoogle Scholar