Dimension groups associated to $\beta$-expansions

Abstract

Completing work by Shultz on one hand and by Katayama, Matsumoto, and Watatani on the other, we prove that a priori different dimension groups associated to β-expansions in fact coincide.

A note on the nuclear dimension of Cuntz–Pimsner -algebras associated with minimal shift spaces

“…This interplay between shift spaces and -algebras starts from the study of the -algebra of a two-sided shift of finite type represented by a -matrix A in a canonical way (see [11]), in which the associated -algebra is originally called a Cuntz–Krieger algebra . In the next 30 years, the -algebra , to every shift space X , is constructed and studied in [1, 5, 7–9, 13, 14, 16–18] by several authors (for example, Matsumoto, Eilers, Carlsen, Brix, and their collaborators, to name a few), but in different manners for their own uses. We additionally remark that the associated -algebra considered in the paper is first defined by Carlsen in [7] using a Cuntz–Pimsner construction, which is why we call it a Cuntz–Pimsner -algebra , as is also pointed out in [4].…”