Self-similar graphs, a unified treatment of Katsura and Nekrashevych C*-algebras

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• 作者：Ruy Exel^a ; ^{ruyexel@gmail.com} ; Enrique Pardo^b ; ^{enrique.pardo@uca.es}
• 关键词：46L05 ; 46L55
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：14 January 2017
• 年：2017
• 卷：306
• 期：Complete
• 页码：1046-1129
• 全文大小：1068 K
• 卷排序：306

文摘

Given a graph E, an action of a group G on E, and a G-valued cocycle φ on the edges of E  , we define a C*-algebra denoted OG,EOG,E, which is shown to be isomorphic to the tight C*-algebra associated to a certain inverse semigroup SG,ESG,E built naturally from the triple (G,E,φ)(G,E,φ). As a tight C*-algebra, OG,EOG,E is also isomorphic to the full C*-algebra of a naturally occurring groupoid Gtight(SG,E)Gtight(SG,E). We then study the relationship between properties of the action, of the groupoid and of the C*-algebra, with an emphasis on situations in which OG,EOG,E is a Kirchberg algebra. Our main applications are to Katsura algebras and to certain algebras constructed by Nekrashevych from self-similar groups. These two classes of C*-algebras are shown to be special cases of our OG,EOG,E, and many of their known properties are shown to follow from our general theory.