Topological entropy and AF subalgebras of graph C*-algebras

Let A(E) be the canonical AF subalgebra of a graph C*-algebra C*(E) associated with a locally finite directed graph E. For Brown and Voiculescu's topological entropy ht(Phi(E)) of the canonical completely positive map Phi(E) on C*(E), ht(Phi(E)) = ht(Phi(E)|(AE)) = h(l)(E) = h(b)(E) is known to hold for a finite graph E, where h(l)(E) is the loop entropy of Gurevic and h(b)(E) is the block entropy of Salama. For an irreducible infinite graph E, the inequality h(l)(E) <= ht(Phi(E)|(AE)) has recently been known. It is shown in this paper that ht(Phi(E)|(AE)) <= max{h(b)(E), h(b)(E-t)}, where E-t is the graph E with the direction of the edges reversed. Some irreducible infinite graphs E-p (p > 1) with ht(Phi(E)|(AEp)) = log p are also examined.