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Journal of Operator Theory

Volume 78, Issue 1,  Summer  2017  pp. 201-225.

Twisted topological graph algebras are twisted groupoid $C^*$-algebras

Authors:  Alex Kumjian (1) and Hui Li (2)
Author institution: (1) Department of Mathematics, Univ. of Nevada (084), Reno, NV 89557, U.S.A.
(2) Research Center for Operator Algebras and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, Department of Mathematics, East China Normal University, 3663 Zhongshan North Road, Putuo District, Shanghai, 200062, China

Summary:  In \textit{Twisted topological graph algebras}, to appear in Houston J. Math., the second author showed how Katsura's construction of the $C^*$-algebra of a topological graph $E$ may be twisted by a Hermitian line bundle $L$ over the edge space $E^1$. The correspondence defining the algebra is obtained as the completion of the compactly supported continuous sections of $L$. We prove that the resulting $C^*$-algebra is isomorphic to a twisted groupoid $C^*$-algebra where the underlying groupoid is the Renault--Deaconu groupoid of the topological graph with Yeend's boundary path space as its unit space.

Keywords:  $C^*$-algebra, topological graph, principal circle bundle, twisted topological graph algebra, Renault-Deaconu groupoid, twisted groupoid $C^*$-algebra

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