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Journal of Operator Theory

Volume 77, Issue 1, Winter 2017  pp. 217-241.

Topological construction of $C^*$-correspondences for groupoid $C^*$-algebras

Authors:  Rohit Dilip Holkar
Author institution: Department of Mathematics, Federal University of Santa Catarina, Florianopolis, 88.040-900, Brazil

Summary: Let $(G,\alpha)$ and $(H,\beta)$ be locally compact groupoids with Haar systems. We define a topological correspondence from $(G,\alpha)$ to $(H,\beta)$ to be a $G$-$H$-bispace $X$ on which $H$ acts properly, and $X$ carries a continuous family of measures which is $H$-invariant and each measure in the family is $(G,\alpha)$-quasi invariant. We show that a topological correspondence produces a $C^*$-correspondence from $C^*(G,\alpha)$ to $C^*(H,\beta)$. We give many examples of topological correspondences.

Keywords: topological correspondences, morphisms of groupoids, groupoid correspondences

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