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The Category of Ordered Bratteli Diagrams

Part of: Ergodic theory Methods of category theory in functional analysis Topological dynamics $K$-theory and operator algebras

Published online by Cambridge University Press:  03 September 2019

Massoud Amini ,
George A. Elliott  and
Nasser Golestani
Massoud Amini
Affiliation:
Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5746, Iran Email: mamini@modares.ac.irmamini@ipm.ir
George A. Elliott
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4 Email: elliott@math.toronto.edu
Nasser Golestani
Affiliation:
Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran Email: n.golestani@modares.ac.ir
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Abstract

A category structure for ordered Bratteli diagrams is proposed in which isomorphism coincides with the notion of equivalence of Herman, Putnam, and Skau. It is shown that the natural one-to-one correspondence between the category of Cantor minimal systems and the category of simple properly ordered Bratteli diagrams is in fact an equivalence of categories. This gives a Bratteli–Vershik model for factor maps between Cantor minimal systems. We give a construction of factor maps between Cantor minimal systems in terms of suitable maps (called premorphisms) between the corresponding ordered Bratteli diagrams, and we show that every factor map between two Cantor minimal systems is obtained in this way. Moreover, solving a natural question, we are able to characterize Glasner and Weiss’s notion of weak orbit equivalence of Cantor minimal systems in terms of the corresponding C*-algebra crossed products.

Keywords

Cantor minimal systemordered Bratteli diagrampremorphismcategoryfunctorC*-algebradimension groupweak orbit equivalence

MSC classification

Primary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Secondary: 46M15: Categories, functors 37A20: Orbit equivalence, cocycles, ergodic equivalence relations 19K14: $K_0$ as an ordered group, traces
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 1 , February 2021 , pp. 1 - 28
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

The first and third authors were in part supported by a grant from IPM (Nos. 96430215 and 94470072). The research of the second author was supported by the Natural Sciences and Engineering Research Council of Canada, and by the Fields Institute. The research of the second author was supported by the Natural Sciences and Engineering Research Council of Canada.

References

Amini, M., Elliott, G. A., and Golestani, N., The category of Bratteli diagrams. Canad. J. Math. 67(2015), 990–1023. https://doi.org/10.4153/CJM-2015-001-8CrossRefGoogle Scholar
Auslander, J., Minimal flows and their extensions. North-Holland Mathematics Studies, 153; Notas de Matemática [Mathematical Notes], 122, North-Holland Publishing Co., Amsterdam, 1988.Google Scholar
Bezuglyi, S. and Karpel, O., Bratteli diagrams: structure, measures, dynamics. In: Dynamics and numbers. Contemp. Math., 669, Amer. Math. Soc., Providence, RI, 2016, pp. 1–36. https://doi.org/10.1090/conm/669/13421Google Scholar
Bezuglyi, S., Kwiatkowski, J., and Medynets, K., Aperiodic substitution systems and their Bratteli diagrams. Ergodic Theory Dynam. Systems 29(2009), no. 1, 37–72. https://doi.org/10.1017/S0143385708000230CrossRefGoogle Scholar
Bratteli, O., Inductive limits of finite dimensional C*-algebras. Trans. Amer. Math. Soc. 171(1972), 195–234. https://doi.org/10.2307/1996380Google Scholar
Dadarlat, M., Morphisms of simple tracially AF algebras. Internat. J. Math. 15(2004), 919–957. https://doi.org/10.1142/S0129167X04002636CrossRefGoogle Scholar
Downarowicz, T., Survey of odometers and Toeplitz flows. In: Algebraic and topological dynamics. Contemp. Math., 385, American Mathematical Society, Providence, RI, 2005, pp. 7–37. https://doi.org/10.1090/conm/385/07188CrossRefGoogle Scholar
Durand, F., Combinatorics on Bratteli diagrams and dynamical systems. In: Combinatorics, automata and number theory. Encyclopedia Math. Appl., 135, Cambridge Univ. Press, Cambridge, 2010, pp. 324–372.CrossRefGoogle Scholar
Durand, F. and Downarowicz, T., Factors of Toeplitz flows and other almost 1–1 extensions over group rotations. Math. Scand. 90(2002), 57–72. https://doi.org/10.7146/math.scand.a-14361Google Scholar
Durand, F., Host, B., and Skau, C. F., Substitutional dynamical systems, Bratteli diagrams and dimension groups. Ergodic Theory Dynam. Systems 19(1999), 953–993. https://doi.org/10.1017/S0143385799133947CrossRefGoogle Scholar
Elliott, G. A., On the classification of inductive limits of sequences of semisimple finite dimensional algebras. J. Algebra 38(1976), 29–44. https://doi.org/10.1016/0021-8693(76)90242-8CrossRefGoogle Scholar
Elliott, G. A., Towards a theory of classification. Adv. Math. 223(2010), 30–48. https://doi.org/10.1016/j.aim.2009.07.018CrossRefGoogle Scholar
Giordano, R., Putnam, I. F., and Skau, C. F., K-theory and asymptotic index for certain almost one-to-one factors. Math. Scand. 89(2001), 297–319. https://doi.org/10.7146/math.scand.a-14343CrossRefGoogle Scholar
Giordano, R., Putnam, I. F., and Skau, C. F., Topological orbit equivalence and C*-crossed products. J. Reine Angew. Math. 469(1995), 51–111.Google Scholar
Gjerde, R. and Johansen, O., Bratteli–Vershik models for Cantor minimal systems: applications to Toeplitz flows. Ergodic Theory Dynam. Systems 20(2000), 1687–1710. https://doi.org/10.1017/S0143385700000948CrossRefGoogle Scholar
Glasner, E. and Host, B., Extensions of Cantor minimal systems and dimension groups. J. Reine Angew. Math. 682(2013), 207–243.Google Scholar
Glasner, E. and Weiss, B., Weak orbit equivalence of Cantor minimal systems. Internat. J. Math. 6(1995), 559–579. https://doi.org/10.1142/S0129167X95000213CrossRefGoogle Scholar
Grothendieck, A., Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats. In: Séminaire Bourbaki, Vol. 5, Exp. No. 190. Société Mathématique de France, Paris, 1995, pp. 299–327.Google Scholar
Golestani, N. and Hosseini, M., Factors of Cantor minimal systems, in preparation.Google Scholar
Herman, R. H., Putnam, I. F., and Skau, C. F., Ordered Bratteli diagrams, dimension groups and topological dynamics. Internat. J. Math. 3(1992), 827–864. https://doi.org/10.1142/S0129167X92000382CrossRefGoogle Scholar
Lin, H., Tracial equivalence for C*-algebras and orbit equivalence for minimal dynamical systems. Proc. Edinb. Math. Soc. (Series 2) 48(2005), 673–690. https://doi.org/10.1017/S0013091503000889CrossRefGoogle Scholar
Mac Lane, S., Categories for the working mathematician, Second ed., Graduate Texts in Mathematics, 5, Springer-Verlag, New York, 1998.Google Scholar
Matui, H., Topological orbit equivalence of locally compact Cantor minimal systems. Ergodic Theory and Dynam. Systems 22(2002), 1871–1903. https://doi.org/10.1017/S0143385702000688CrossRefGoogle Scholar
Power, S. C., Algebraic orders on K 0 and approximately finite operator algebras. J. Operator Theory 27(1992), no. 1, 87–106.Google Scholar
Putnam, I. F., C*-algebras associated with minimal homeomorphisms of the Cantor set. Pacific J. Math. 136(1989), 329–353.CrossRefGoogle Scholar
Putnam, I. F., Cantor minimal systems, Vol. 70. American Mathematical Soc., 2018.CrossRefGoogle Scholar
Rørdam, M., Classification of nuclear, simple C*-algebras. In: Classification of nuclear C*-algebras. Entropy in operator algebras. Encyclopaedia of Mathematical Sciences, 126, Oper. Alg. Non-commut. Geom., 7, Springer, Berlin, 2002, pp. 1–145. https://doi.org/10.1007/978-3-662-04825-2_1CrossRefGoogle Scholar
Sugisaki, F., Almost one-to-one extensions of Cantor minimal systems and order embeddings of simple dimension groups. Münster J. Math. 4(2011), 141–169.Google Scholar
Vershik, A. M., Uniform algebraic approximations of shift and multiplication operators. Dokl. Akad. Nauk SSSR 259(1981), 526–529. English translation: Sov. Math. Dokl. 24(1981), 97–100.Google Scholar
Vershik, A. M., A theorem on periodical Markov approximation in ergodic theory. In: Ergodic theory and related topics (Vitte, 1981). Math. Res., 12, Akademie-Verlag, Berlin, 1982, pp. 195–206.Google Scholar