## Rocky Mountain Journal of Mathematics

### C*-algebras of 2-groupoids

Massoud Amini

#### 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

#### 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

#### References

• M. Amini, Tannaka-Krein duality for compact groupoids I, Representation theory, Adv. Math. 214 (2007), 78–91.
• ––––, Tannaka-Krein duality for compact groupoids II, Duality, Oper. Matr. 4 (2010), 573–592.
• J.C. Baez, An introduction to $n$-categories, in Category theory and computer science, Lect. Notes Comp. Sci. 1290, Springer, Berlin, 1997.
• N. Bourbaki, Integration, Herman, Paris, 1963.
• R. Brown and K.C.H. Mackenzie, Determination of a double Lie groupoid by its core diagram, J. Pure Appl. Alg. 80 (1992), 237–272.
• M.R. Buneci, Isomorphic groupoid $C^*$-algebras associated with different Haar systems, New York J. Math. 11 (2005), 225–245.
• A. Buss, R. Meyer and C. Zhu, Non-Hausdorff symmetries of (C^*)-algebras, Math. Ann. 352 (2012), 73–97.
• ––––, A higher category approach to twisted actions on (C^*)-algebras, Proc. Edinb. Math. Soc. 56 (2013), 387–426.
• A.M. Cegarra and J. Remedios, The relationship between the diagonal and the bar constructions on a bisimplicial set, Topol. Appl. 153 (2005), 21–51.
• A. Connes, A survey of foliations and operator algebras, in Operator algebras and applications, Part I, American Mathematical Society, Providence, R.I., 1982.
• J.M.G. Fell, Weak containment and induced representations of groups, Canad. J. Math. 14 (1962), 237–268.
• Philip Green, The local structure of twisted covariance algebras, Acta Math. 140 (1978), 191–250.
• A. Guichardet, Une caractèrisation des algèbres de von Neumann discrètes, Bull. Soc. Math. France 89 (1961), 77–101.
• A. Haefliger, Orbi-espaces, in Sur les Groupes Hyperboliques d'après Mikhael Gromov, Prog. Math. 83, Birkhäuser, Boston, 1990.
• P. Hahn, Haar measure and convolution algebra on ergodic groupoids, Ph.D. thesis, Harvard University, Cambridge, MA, 1975.
• Z.J. Liu, A. Weinstein and P. Xu, Manin triples for Lie bialgebroids, J. Diff. Geom. 45 (1997), 547–574.
• J.-H. Lu and A. Weinstein, Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Diff. Geom. 31 (1990), 501–526.
• K.C.H. Mackenzie, Double Lie algebroids and second-order geometry, I, Adv. Math. 94 (1992), 180–239.
• R.A. Mehta and X. Tang, From double Lie groupoids to local Lie $2$-groupoids, Bull. Brazilian Math. Soc. 42 (2011), 651–681.
• P.S. Muhly, Bundles over groupoids, in Groupoids in analysis, geometry, and physics American Mathematical Society, Providence, RI, 2001.
• B. Noohi, Notes on $2$-groupoids, $2$-groups and crossed modules, Homol. Homot. Appl. 9 (2007), 75–106.
• A.L.T. Paterson, Groupoids, inverse semigroups, and their operators algebras, Progr. Math. 170, Birkhäuser, Boston, 1999.
• A. Ramsay, Virtual groups and group actions, Adv. Math. 6 (1971), 253–322.
• J. Renault, A groupoid approach to $C^*$-algebras, Lect. Notes Math. 793, Springer-Verlag, Berlin, 1980.
• M. Rieffel, Induced representations of $C^*$-algebras, Adv. Math. 13 (1974), 176–257.
• L. Stefanini, On the integration of LA-groupoids and duality for Poisson groupoids, Trav. Math. 17 (2007), 39–59.
• A. Weinstein, Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan 40 (1988), 705–727.
• P. Xu, Momentum maps and Morita equivalence, J. Diff. Geom. 67 (2004), 289–333.
• C. Zhu, Kan replacement of simplicial manifolds, Lett. Math. Phys. 90 (2009), 383–405.