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CUNTZ–KRIEGER ALGEBRAS ASSOCIATED TO SELF-SIMILAR GROUPOIDS
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 110 / Issue 3 / December 2024
- Published online by Cambridge University Press:
- 27 September 2024, pp. 568-569
- Print publication:
- December 2024
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- Article
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CUNTZ–KRIEGER ALGEBRAS AND ONE-SIDED CONJUGACY OF SHIFTS OF FINITE TYPE AND THEIR GROUPOIDS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 109 / Issue 3 / December 2020
- Published online by Cambridge University Press:
- 21 June 2019, pp. 289-298
- Print publication:
- December 2020
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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.
