Functoriality of the bimodule associated to a Hilsum-Skandalis map. (English) Zbl 0938.22002

The author defines a functor from the category of smooth functors between separated etale groupoids to the category of bimodules over locally unital algebras. More precisely, given two etale groupoids \(G\) and \(H\) he considers the isomorphism class of a principal \(G\)-\(H\)-bimodule \(E\), which makes up the Hilsum-Skandalis maps, and to this he associates the isomorphism class of the \(C_c^\infty(G)\)-\(C_c^\infty(H)\)-bimodule \(C_c^\infty(E)\). As an immediate consequence of this construction the author deduces that Morita equivalent etale groupoids \(G\) and \(H\) give rise to Morita equivalent convolution algebras \(C_c^\infty(G)\) and \(C_c^\infty(H)\).


22A22 Topological groupoids (including differentiable and Lie groupoids)
58H05 Pseudogroups and differentiable groupoids
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