Finite simple labeled graph C-algebras of Cantor minimal subshifts

It is well known that a simple graph C*-algebra is either AF or purely infinite. In this paper, we address the question of whether this is the case for labeled graph C*-algebras which include all graph C*-algebras and Matsumoto algebras of subshifts. There have been various C*-algebra constructions associated with subshifts and some of them are known to have the crossed products C(X) x(T) Z of Cantor minimal subshifts (X, T) as their quotient algebras. We show that such a simple crossed product C(X) X-T Z can be realised as a labeled graph C*-algebra. Since this C*-algebra is known to be an AT algebra and has Z as its K-1-group, our result provides a family of simple finite non-AF unital labeled graph C*-algebras. (C) 2016 Elsevier Inc. All rights reserved.