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CUNTZ–KRIEGER ALGEBRAS AND ONE-SIDED CONJUGACY OF SHIFTS OF FINITE TYPE AND THEIR GROUPOIDS

Part of: Selfadjoint operator algebras Ergodic theory Topological dynamics

Published online by Cambridge University Press:  21 June 2019

KEVIN AGUYAR BRIX Open the ORCID record for KEVIN AGUYAR BRIX [Opens in a new window]  and
TOKE MEIER CARLSEN Open the ORCID record for TOKE MEIER CARLSEN [Opens in a new window]
KEVIN AGUYAR BRIX*
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100Copenhagen, Denmark
TOKE MEIER CARLSEN
Affiliation:
Department of Science and Technology, University of the Faroe Islands, Vestara Bryggja 15, FO-100Tórshavn, The Faroe Islands e-mail: toke.carlsen@gmail.com
*
e-mail: kab@math.ku.dk
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Abstract

A one-sided shift of finite type $(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$ determines on the one hand a Cuntz–Krieger algebra ${\mathcal{O}}_{A}$ with a distinguished abelian subalgebra ${\mathcal{D}}_{A}$ and a certain completely positive map $\unicode[STIX]{x1D70F}_{A}$ on ${\mathcal{O}}_{A}$. On the other hand, $(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$ determines a groupoid ${\mathcal{G}}_{A}$ together with a certain homomorphism $\unicode[STIX]{x1D716}_{A}$ on ${\mathcal{G}}_{A}$. We show that each of these two sets of data completely characterizes the one-sided conjugacy class of $\mathsf{X}_{A}$. This strengthens a result of Cuntz and Krieger. We also exhibit an example of two irreducible shifts of finite type which are eventually conjugate but not conjugate. This provides a negative answer to a question of Matsumoto of whether eventual conjugacy implies conjugacy.

Keywords

Shifts of finite typegroupoidsCuntz–Krieger algebras

MSC classification

Primary: 46L55: Noncommutative dynamical systems
Secondary: 37A55: Relations with the theory of $C^*$-algebras 37B10: Symbolic dynamics
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 109 , Issue 3 , December 2020 , pp. 289 - 298
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by L. O. Clarke

The first named author is supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).

References

Boyle, M., Fiebig, D. and Fiebig, U., ‘A dimension group for local homeomorphisms and endomorphisms of onesided shifts of finite type’, J. reine angew. Math. 487 (1997), 27–59.Google Scholar
Carlsen, T. M., Eilers, S., Ortega, E. and Restorff, G., ‘Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids’, J. Math. Anal. Appl. 469 (2019), 1088–1110.Google Scholar
Carlsen, T. M. and Rout, J., ‘Diagonal-preserving gauge-invariant isomorphisms of graph C ∗ -algebras’, J. Funct. Anal. 273 (2017), 2981–2993.Google Scholar
Carlsen, T. M., Ruiz, E., Sims, A. and Tomforde, M., ‘Reconstruction of groupoids and $C^{\ast }$ -rigidity of dynamical systems’, Preprint, 2017, arXiv:1711.01052, 44 pages.Google Scholar
Cuntz, J., ‘A class of C ∗-algebras and topological Markov chains. II. Reducible chains and the Ext-functor for C ∗-algebras’, Invent. Math. 63 (1981), 25–40.Google Scholar
Cuntz, J. and Krieger, W., ‘A class of C ∗-algebras and topological Markov chains’, Invent. Math. 56 (1980), 251–268.Google Scholar
Deaconu, V., ‘Groupoids associated with endomorphisms’, Trans. Amer. Math. Soc. 347 (1995), 1779–1786.Google Scholar
Kitchens, B. P., Symbolic Dynamics, One-Sided, Two-Sided and Countable State Markov Shifts, Universitext (Springer, Berlin, 1998).Google Scholar
Kumjian, A., Pask, D., Raeburn, I. and Renault, J., ‘Graphs, groupoids, and Cuntz–Krieger algebras’, J. Funct. Anal. 144 (1997), 505–541.Google Scholar
Lind, D. and Marcus, B., An Introduction to Symbolic Dynamics and Coding (Cambridge University Press, Cambridge, 1995).Google Scholar
Matsumoto, K., ‘Orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras’, Pacific J. Math. 246 (2010), 199–225.Google Scholar
Matsumoto, K., ‘Continuous orbit equivalence, flow equivalence of Markov shifts and circle actions on Cuntz–Krieger algebras’, Math. Z. 285 (2017), 121–141.Google Scholar
Matsumoto, K., ‘Uniformly continuous orbit equivalence of Markov shifts and gauge actions on Cuntz–Krieger algebras’, Proc. Amer. Math. Soc. 145 (2017), 1131–1140.Google Scholar
Matsumoto, K. and Matui, H., ‘Continuous orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras’, Kyoto J. Math. 54 (2014), 863–877.Google Scholar
Matui, H., ‘Homology and topological full groups of étale groupoids on totally disconnected spaces’, Proc. Lond. Math. Soc. (3) 104(1) (2012), 27–56.Google Scholar
Renault, J., A Groupoid Approach to C ∗-Algebras, Lecture Notes in Mathematics, 793 (Springer, Berlin, 1980).Google Scholar
Renault, J., ‘Cartan subalgebras in C ∗-algebras’, Irish Math. Soc. Bull. 61 (2008), 29–63.Google Scholar
Sims, A., ‘Hausdorff étale groupoids and their C ∗-algebras to appear in the volume Operator algebras and dynamics: groupoids, crossed products and Rokhlin dimension’, in: Advanced Courses in Mathematics. CRM Barcelona (Birkäuser, Basel) (arXiv:1710.10897 [math.OA]).Google Scholar
Sims, A. and Williams, D. P., ‘The primitive ideals of some étale groupoid C ∗ -algebras’, Algebr. Represent. Theory 19 (2016), 255–276.Google Scholar