Finite simple labeled graph C⁎-algebras of Cantor minimal subshifts

Abstract 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 ) × T Z of Cantor minimal subshifts ( X , T ) as their quotient algebras. We show that such a simple crossed product C ( X ) × T Z can be realized as a labeled graph C ⁎ -algebra. Since this C ⁎ -algebra is known to be an A T algebra and has Z as its K 1 -group, our result provides a family of simple finite non-AF unital labeled graph C ⁎ -algebras.