The classification of two-component Cuntz-Krieger algebras

Huang, Danrun

doi:10.1090/S0002-9939-96-03079-1

The classification of two-component Cuntz-Krieger algebras
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by Danrun Huang

Proc. Amer. Math. Soc. 124 (1996), 505-512

DOI: https://doi.org/10.1090/S0002-9939-96-03079-1

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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

Mike Boyle, Symbolic dynamics and matrices, Combinatorial and graph-theoretical problems in linear algebra (Minneapolis, MN, 1991) IMA Vol. Math. Appl., vol. 50, Springer, New York, 1993, pp. 1–38. MR 1240955, DOI 10.1007/978-1-4613-8354-3_{1}
Rufus Bowen and John Franks, Homology for zero-dimensional nonwandering sets, Ann. of Math. (2) 106 (1977), no. 1, 73–92. MR 458492, DOI 10.2307/1971159
Joachim Cuntz and Wolfgang Krieger, A class of $C^{\ast }$-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268. MR 561974, DOI 10.1007/BF01390048
J. Cuntz, The classification problem for the $C^\ast$-algebras ${\scr O}_A$, Geometric methods in operator algebras (Kyoto, 1983) Pitman Res. Notes Math. Ser., vol. 123, Longman Sci. Tech., Harlow, 1986, pp. 145–151. MR 866492
—, Personal communication.
J. Cuntz, On the homotopy groups of the space of endomorphisms of a $C^{\ast }$-algebra (with applications to topological Markov chains), Operator algebras and group representations, Vol. I (Neptun, 1980) Monogr. Stud. Math., vol. 17, Pitman, Boston, MA, 1984, pp. 124–137. MR 731768
Joachim Cuntz and Wolfgang Krieger, A class of $C^{\ast }$-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268. MR 561974, DOI 10.1007/BF01390048
John Franks, Flow equivalence of subshifts of finite type, Ergodic Theory Dynam. Systems 4 (1984), no. 1, 53–66. MR 758893, DOI 10.1017/S0143385700002261
D. Huang, Flow equivalence of reducible shifts of finite type, Ergodic Theory Dynamical Systems 14 (1994), 695–720, .
W. Parry and D. Sullivan, A topological invariant for flows on one dimensional spaces, Topology 14 (1975), 297–299.
M. Rørdam, Classification of Cuntz-Krieger algebras, $K$-theory 9 (1995), 31–58.