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Journal of Operator Theory

Volume 56, Issue 1, Summer 2006  pp. 199-217.

Actions and coactions of measured groupoids on $W^*$-algebras

Authors:  Karla J. Oty (1) and Arlan Ramsay (2)
Author institution: (1) Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA
(2) Department of Mathematics, University of Colorado, Boulder, CO 80309, USA

Summary:  This paper answers some questions involved in extending from groups to groupoids the theory of actions and coactions on $W^*$-algebras. In particular, we explain the connection between actions of a measured groupoid, $G$, on a bundle of $W^*$-algebras, and Hopf actions of the Hopf algebroid $L^\infty(G)$ on the direct integral of the bundle of $W^*$-algebras. The Hopf algebroid structure on $L^\infty(G)$ is determined by $G$ and can be used to construct $G$ up to a set of measure zero.

Keywords:  groupoids, $W^*$-algebras, actions, coactions, duality

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