Topological entropy and AF subalgebras of graph 𝐶*-algebras

Jeong, Ja; Park, Gi

doi:10.1090/S0002-9939-05-08165-7

Topological entropy and AF subalgebras of graph $C^*$-algebras
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Proc. Amer. Math. Soc. 134 (2006), 215-228 Request permission

Abstract:

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 $C^*(E)$, $ht(\Phi _E)=ht(\Phi _E|_{\mathcal A_E})=h_l(E)=h_b(E)$ 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 $h_l(E)\leq ht(\Phi _E|_{\mathcal A_E})$ has recently been known. It is shown in this paper that \[ ht(\Phi _E|_{\mathcal A_E})\leq \max \{ h_b(E), h_b( {}^{t}E)\},\] where ${}^tE$ is the graph $E$ with the direction of the edges reversed. Some irreducible infinite graphs $E_p\ (p>1)$ with $ht(\Phi _E|_{\mathcal A_{E_p}})=\log p$ are also examined.

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