C*-algebras of 2-groupoids

Abstract

We define topological 2-groupoids and study locally compact 2-groupoids with 2-Haar systems. We consider quasi-invariant measures on the sets of 1-arrows and unit space and build the corresponding vertical and horizontal modular functions. For a given 2-Haar system, we construct the vertical and horizontal full \Cst -algebras of a 2-groupoid and show that they are independent of the choice of the 2-Haar system, up to strong Morita equivalence. We make a correspondence between their bounded representations on Hilbert spaces and those of the 2-groupoid on Hilbert bundles. We show that representations of certain closed 2-subgroupoids are induced to representations of the 2-groupoid and use regular representation to build the vertical and horizontal reduced \Cst -algebras of the 2-groupoid. We establish strong Morita equivalence between \Cst -algebras of the 2-groupoid of composable pairs and those of the 1-arrows and unit space. We describe the reduced \Cst -algebras of r-discrete principal 2-groupoids and find their ideals and masa's.