Cuntz-Krieger algebras and one-sided conjugacy of shifts of finite type and their groupoids

Abstract

A one-sided shift of finite type $(X_A,{\sigma }_A)$ 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 ${\tau }_A$ on ${\mathcal{O}}_A$. On the other hand, $(X_A,{\sigma }_A)$ determines a groupoid ${\mathcal{G}}_A$ together with a certain homomorphism ${\epsilon }_A$ on ${\mathcal{G}}_A$. We show that this data completely characterizes the one-sided conjugacy class of $X_A$. 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 answers a question of Matsumoto of whether eventual conjugacy implies conjugacy in the negative.

v2: Minor changes, corollary added, references updated; 9 pages. This is the version that will be published

References in corpus (4)

• Continuous orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras
• Homology and topological full groups of etale groupoids on totally disconnected spaces
• Diagonal-preserving gauge-invariant isomorphisms of graph $C^{\ast }$-algebras
• Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids

Cited by in corpus (2)

• Conjugacy of local homeomorphisms via groupoids and C*-algebras
• C*-algebras, groupoids and covers of shift spaces