Journal of Operator Theory

Volume 46, Issue 2, Fall 2001 pp. 251-264.

Cohomology of topological graphs and Cuntz-Pimsner algebras

Authors: V. Deaconu (1), A. Kumjian (2), and P. Muhly (3)
Author institution: (1) Department of Mathematics/084, University of Nevada, Reno, NV 89557, USA
(2) Department of Mathematics/084, University of Nevada, Reno, NV 89557, USA
(3) Department of Mathematics, Universiy of Iowa, Iowa City, IA 52242, USA

Summary: We compute the sheaf cohomology of a groupoid built from a local homeomorphism of a locally compact space $X$. In particular, we identify the twists over this groupoid, and its Brauer group. Our calculations refine those made by Kumjian, Muhly, Renault and Williams in the case $X$ is the path space of a graph, and the local homeomorphism is the shift. We also show how the $C^*$-algebra of a twist may be identified with the Cuntz-Pimsner algebra constructed from a certain $C^*$-correspondence.

Keywords: Groupoid cohomology, Cuntz-Pimsner algebras, Hilbert modules, Hilbert bimodules, and $C^*$-correspondence

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