Actions of 2-groupoids on C*-algebras

• Applicant: Professor Dr. Ralf Meyer
• Subject Area: Mathematics
• Term: from 2009 to 2014
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 128675010

• Final Report Year: 2016

Final Report Abstract

This project studied a notion of symmetry for noncommutative spaces, where groups are replaced by 2-groups or crossed modules. These have an extra layer of structure, 2-arrows between arrows, which express the possibility that different arrows can be equivalent. Both groupoids and operator algebras naturally have such kinds of symmetries because there is a normal subgroup of automorphisms – the inner ones – that are more trivial than generic automorphisms. This more general notion of symmetry allows more general notions of dynamical systems. In particular, crossed modules may act on C∗-algebras. A crossed product encodes such an action in a single C∗-algebra. The structure of crossed products for crossed module actions has been understood very well by reducing their construction to known ones like crossed products for group actions and taking fibres in fields of C∗-algebras. The project has also defined actions of possibly non-Hausdorff, étale groupoids on other groupoids by equivalences, in such a way that they lead to actions on C∗-algebras as well. The initial hope that this could help to prove the Baum–Connes conjecture for large classes of non-Hausdorff groupoids could not yet be realised.

Publications

• Actions of higher categories on C*-algebras, 3rd Annual Luis Santaló Winter School–CIMPA Research School (Universidad de Buenos Aires, July 26), Topics in non commutative geometry (Guillermo Cortiñas, ed.), Clay Mathematics Proceedings, vol. 16, Amer. Math. Soc., Providence, RI, 2012
  Ralf Meyer
  
• Non-Hausdorff symmetries of C∗-algebras, Math. Ann. 352 (2012), no. 1, 73–97
  Alcides Buss, Ralf Meyer, and Chenchang Zhu
  (See online at https://doi.org/10.1007/s00208-010-0630-3)
• Strictification of étale stacky Lie groups, Compos. Math. 148 (2012), no. 3, 807–834
  Giorgio Trentinaglia and Chenchang Zhu
  (See online at https://doi.org/10.1112/S0010437X11007020)
• A higher category approach to twisted actions on C∗ -algebras, Proc. Edinb. Math. Soc. (2) 56 (2013), no. 2, 387–426
  Alcides Buss, Ralf Meyer, and Chenchang Zhu
  (See online at https://doi.org/10.1017/S0013091512000259)
• Higher groupoid actions, bibundles, and differentiation, Ph.D. Thesis, Georg-August-Universität Göttingen, 2014
  Du Li
  
• Topological construction of C∗-correspondences for groupoid C∗-algebras, Ph.D. Thesis, Georg-August-Universität Göttingen, 2014
  Rohit Dilip Holkar
  
• Groupoids in categories with pretopology, Theory Appl. Categ. 30 (2015), 1906–1998
  Ralf Meyer and Chenchang Zhu
  
• Crossed products for actions of crossed modules on C∗ -algebras, J. Noncommut. Geom. (2016)
  Alcides Buss and Ralf Meyer
  
• Inverse semigroup actions on groupoids, Rocky Mountain J. Math. (2016)
  Alcides Buss and Ralf Meyer
  
• Reduced C ∗ -algebras of Fell bundles over inverse semigroups, Israel J. Math. (2016)
  Alcides Buss, Ruy Exel, and Ralf Meyer