Finite C*-algebras associated with labeled graphs

Abstract

We study the properties of the C*-algebras C*(E,L,B) associated to labeled spaces (E,L,B). It is shown that if C*(E,L,B) is AF, then the labeled space (E,L,B) has no loops. We also prove that some of the known equivalent conditions for usual graph C*-algebras C*(E) to be AF are not necessarily equivalent for labeled graph C*-algebras by providing examples. For this, we use generalized Morse sequences. These examples are also shown to be non-AF simple finite C*-algebras, which contrasts with the fact that the usual simple graph C*-algebras are either AF or purely infinite.

Besides, we find a sufficient condition for a labeled space (E,L,B) to give rise to an infinite C*-algebra in the sense that every nonzero hereditary C*-subalgebra of C*(E,L,B) contains an infinite projection.