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

Abstract

AbstractA 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.