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Purely infinite labeled graph $C^{\ast }$-algebras

Published online by Cambridge University Press:  04 December 2017

JA A JEONG ,
EUN JI KANG  and
GI HYUN PARK
JA A JEONG
Affiliation:
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea email jajeong@snu.ac.kr
EUN JI KANG
Affiliation:
BK21 Plus Mathematical Sciences Division, Seoul National University, Seoul 08826, Korea email kkang33@snu.ac.kr
GI HYUN PARK
Affiliation:
Department of Financial Mathematics, Hanshin University, Osan 18101, Korea email ghpark@hs.ac.kr
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Abstract

In this paper, we consider pure infiniteness of generalized Cuntz–Krieger algebras associated to labeled spaces $(E,{\mathcal{L}},{\mathcal{E}})$. It is shown that a $C^{\ast }$-algebra $C^{\ast }(E,{\mathcal{L}},{\mathcal{E}})$ is purely infinite in the sense that every non-zero hereditary subalgebra contains an infinite projection (we call this property (IH)) if $(E,{\mathcal{L}},{\mathcal{E}})$ is disagreeable and every vertex connects to a loop. We also prove that under the condition analogous to (K) for usual graphs, $C^{\ast }(E,{\mathcal{L}},{\mathcal{E}})=C^{\ast }(p_{A},s_{a})$ is purely infinite in the sense of Kirchberg and Rørdam if and only if every generating projection $p_{A}$, $A\in {\mathcal{E}}$, is properly infinite, and also if and only if every quotient of $C^{\ast }(E,{\mathcal{L}},{\mathcal{E}})$ has property (IH).

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 8 , August 2019 , pp. 2128 - 2158
Copyright
© Cambridge University Press, 2017 

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