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

It is well known that a simple graph C⁎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⁎C⁎-algebras which include all graph C⁎C⁎-algebras and Matsumoto algebras of subshifts. There have been various C⁎C⁎-algebra constructions associated with subshifts and some of them are known to have the crossed products C(X)×TZC(X)×TZ of Cantor minimal subshifts (X,T)(X,T) as their quotient algebras.We show that such a simple crossed product C(X)×TZC(X)×TZ can be realized as a labeled graph C⁎C⁎-algebra. Since this C⁎C⁎-algebra is known to be an ATAT algebra and has ZZ as its K1K1-group, our result provides a family of simple finite non-AF unital labeled graph C⁎C⁎-algebras.