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

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 ${\mathcal{O}}_{A}$...

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Bibliographic Details
Published in:Journal of the Australian Mathematical Society (2001) 2020-12, Vol.109 (3), p.289-298
Main Authors: BRIX, KEVIN AGUYAR, CARLSEN, TOKE MEIER
Format: Article
Language:English
Subjects:
Conjugates
Homomorphisms
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Summary: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.
ISSN:1446-7887
1446-8107
DOI:10.1017/S1446788719000168