Topological entropy and the AF core of a graph C∗-algebra

Let C∗(E) be the C∗-algebra associated with a locally finite directed graph E and AE be the AF core of C∗(E). For the topological entropy ht(ΦE) (in the sense of Brown-Voiculescu) of the canonical completely positive map ΦE on the graph C∗-algebra, it is known that if E is finiteht(ΦE)=ht(ΦE|AE)=hb(E)=hl(E), where hb(E) (respectively, hl(E)) is the block (respectively, the loop) entropy of E. In case E is irreducible and infinite, hl(E)⩽ht(ΦE|AE)⩽hb(Et) is known recently, where Et is the graph E with the edges directed reversely. Then by monotonicity of entropy, hl(E)⩽ht(ΦE) is clear. In this paper we show that ht(ΦE)⩽hb(Et) holds for locally finite infinite graphs E. The AF core AE is known to be stably isomorphic to the graph C∗-algebra C∗(E×cZ) of certain skew product E×cZ and we also show that ht(ΦE×cZ)=ht(ΦE|AE). Examples Ep (p>1) of irreducible graphs with ht(ΦEp)=logp are discussed.