# 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.

**DOI: **http://dx.doi.org/10.7900/jot.2016jun23.2136

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

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