Topological entropy and AF subalgebras of graph $C^*$-algebras

Let $\mathcal A_E$ 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 $C^*(E)$, $ht(\Phi_E)=ht(\Phi_E\vert _{\mathcal A_E})=h_l(E)=h_b(E)$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 $h_l(E)\leq ht(\Phi_E\vert _{\mathcal A_E})$ has recently been known. It is shown in this paper that

where ${}^tE$ is the graph $E$ with the direction of the edges reversed. Some irreducible infinite graphs $E_p (p>1)$ with $ht(\Phi_E\vert _{\mathcal A_{E_p}})=\log p$ are also examined.