Topological entropy and AF subalgebras of graph C*-algebras

Abstract

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-Voiculescu's topological entropy ht(Φ_E) of the canonical completely positive map Φ_E on C*(E), ht(Φ_E)=ht(Φ_E|_{A_E})=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)\leq ht(Φ_E|_{A_E}) has been known recently. It is shown in this paper that ht(Φ_E|_{A_E})\leq max{h_b(E), h_b(tE)}, where tE is the graph E with the direction of the edges reversed. Some irreducible infinite graphs E_p(p>1) with ht(Φ_E|_{A_{E_p}})=log p are also examined.