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The classification
of two-component Cuntz-Krieger algebras

Author: Danrun Huang
Journal: Proc. Amer. Math. Soc. 124 (1996), 505-512
MSC (1991): Primary 46L35, 54H20; Secondary 46L55
MathSciNet review: 1301504
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Abstract: Cuntz-Krieger algebras with exactly one nontrivial closed ideal are classified up to stable isomorphism by the Cuntz invariant. The proof relies on Rørdam's classification of simple Cuntz-Krieger algebras up to stable isomorphism and the author's classification of two-component reducible topological Markov chains up to flow equivalence.

References [Enhancements On Off] (What's this?)

  • Bo M. Boyle, Symbolic dynamics and matrices, Combinatorial and Graph-Theoretic Problems in Linear Algebra, IMA Volumes in Math and Appl. Vol. 50 (R. Brualdi, S. Friedland, and V. Klee, eds.), Springer-Verlag, New York 1993, pp. 1--38, MR 94g:58062.
  • BF R. Bowen and J. Franks, Homology for zero-dimensional basic sets, Ann. of Math. (2) 106 (1977), 73--92, MR 56:16692.
  • C1 J. Cuntz, A class of C*-algebras and topological Markov chains II: reducible chains and the Ext-functor for C*-algebras, Inventiones Math. 63 (1981), 25--40, MR 82f:46073b.
  • C2 ------, The classification problem for the $C^{*}$-algebras $\mathcal{O}_{A}$, Geometric methods in operator algebras (H. Araki and E. G. Effros, eds.), Longman, New York, 1986, pp. 145--151, MR 88a:46081.
  • C3 ------, Personal communication.
  • C4 ------, On the homotopy groups of endomorphisms of a $C^*$-algebra (with applications to topological Markov chains), Proc. of O.A.G.R. Conference in Neptun, Romania, Pitman, 1984, pp. 124--137, MR 86a:46093.
  • CK J. Cuntz and W. Krieger, A class of $C^*$-Algebras and topological Markov chains, Inventiones Math. 56 (1980), 251--268, MR 82f:46073a.
  • F J. Franks, Flow equivalence of subshifts of finite type, Ergodic Theory Dynamical Systems 4 (1984), 53--66, MR 86j:58078.
  • H D. Huang, Flow equivalence of reducible shifts of finite type, Ergodic Theory Dynamical Systems 14 (1994), 695--720, CMP 95:04.
  • PS W. Parry and D. Sullivan, A topological invariant for flows on one dimensional spaces, Topology 14 (1975), 297--299.
  • R M. Rørdam, Classification of Cuntz-Krieger algebras, $K$-theory 9 (1995), 31--58.

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Additional Information

Danrun Huang
Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195-4350

Keywords: Cuntz-Krieger algebra, stable isomorphism, topological Markov chain, flow equivalence
Received by editor(s): June 13, 1994
Received by editor(s) in revised form: August 30, 1994
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society