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 A(E) be the AF core of C*(E). For the topological entropy ht(Phi(E)) (in the sense of Brown-Voiculescu) of the canonical completely positive map Phi(E) on the graph C*-algebra, it is known that if E is finite ht(Phi(E)) = ht(Phi(E)vertical bar(AE)) = h(b)(E) = h(l)(E). where h(b)(E) (respectively, h(l)(E)) is the block (respectively, the loop) entropy of E. In case E is irreducible and infinite, h(l)(E) <= ht(Phi(E)vertical bar(AE)) <= h(b)(E(t)) is known recently, where E(t) is the graph E with the edges directed reversely. Then by monotonicity of entropy, h(l)(E) <= ht(Phi(E)) is clear. In this paper we show that ht(Phi(E)) <= h(b)(E(t)) holds for locally finite infinite graphs E. The AF core A(E) is known to be stably isomorphic to the graph C*-algebra C*(E x(c) Z) of certain skew product E x(c) Z and we also show that ht(Phi(Exc)Z) = ht(Phi(E)vertical bar(AE)). Examples E(p) (p > 1) of irreducible graphs with ht(Phi(Ep)) = log p are discussed. (C) 2009 Elsevier Inc. All rights reserved.