Rocky Mountain Journal of Mathematics

C*-algebras of 2-groupoids

Massoud Amini

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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.

Article information

Source
Rocky Mountain J. Math., Volume 46, Number 3 (2016), 693-728.

Dates
First available in Project Euclid: 7 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1473275759

Digital Object Identifier
doi:10.1216/RMJ-2016-46-3-693

Mathematical Reviews number (MathSciNet)
MR3544832

Zentralblatt MATH identifier
1346.18007

Subjects
Primary: 18D05: Double categories, 2-categories, bicategories and generalizations 46L05: General theory of $C^*$-algebras 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]

Keywords
2-Category 2-groupoid 2-Haar system $C^*$-algebras of 2-groupoids induced representations strong Morita equivalence

Citation

Amini, Massoud. C*-algebras of 2-groupoids. Rocky Mountain J. Math. 46 (2016), no. 3, 693--728. doi:10.1216/RMJ-2016-46-3-693. https://projecteuclid.org/euclid.rmjm/1473275759


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