Open Access
2016 C*-algebras of 2-groupoids
Massoud Amini
Rocky Mountain J. Math. 46(3): 693-728 (2016). DOI: 10.1216/RMJ-2016-46-3-693


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.


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Massoud Amini. "C*-algebras of 2-groupoids." Rocky Mountain J. Math. 46 (3) 693 - 728, 2016.


Published: 2016
First available in Project Euclid: 7 September 2016

zbMATH: 1346.18007
MathSciNet: MR3544832
Digital Object Identifier: 10.1216/RMJ-2016-46-3-693

Primary: 18D05 , 46L05 , 46L55

Keywords: $C^*$-algebras of 2-groupoids , 2-category , 2-groupoid , 2-Haar system , Induced representations , strong Morita equivalence

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium

Vol.46 • No. 3 • 2016
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