Purely infinite labeled graph C^*-algebras

2017/03/05 by Jeong, Ja A, Kang, Eun Ji, Park, Gi Hyun
#37A55 #46L05 #46L55 #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1703.01583

Abstract

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

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