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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