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

Authors: Ja A Jeong and Gi Hyun Park
Journal: Proc. Amer. Math. Soc. 134 (2006), 215-228
MSC (2000): Primary 46L05, 46L55
Published electronically: June 29, 2005
MathSciNet review: 2170561
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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 $C^*(E)$, $ht(\Phi_E)=ht(\Phi_E\vert _{\mathcal A_E})=h_l(E)=h_b(E)$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

\begin{displaymath}ht(\Phi_E\vert _{\mathcal A_E})\leq \max\{ h_b(E), h_b(\,{}^{t}E)\},\end{displaymath}

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.

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Additional Information

Received by editor(s): March 22, 2004
Received by editor(s) in revised form: August 26, 2004
Published electronically: June 29, 2005
Additional Notes: The first author was partially supported by KOSEF R14-2003-006-01000-0
The second author was partially supported by KOSEF R01-2001-000-00001-0
Communicated by: David R. Larson
Article copyright: © Copyright 2005 American Mathematical Society

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