Weakly proper group actions, Mansfield’s Imprimitivity and twisted Landstad duality
Authors:
Alcides Buss and Siegfried Echterhoff
Journal:
Trans. Amer. Math. Soc. 368 (2016), 249-280
MSC (2010):
Primary 46L55, 22D35
DOI:
https://doi.org/10.1090/S0002-9947-2015-06406-X
Published electronically:
March 4, 2015
MathSciNet review:
3413863
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Using the theory of weakly proper actions of locally compact groups recently developed by the authors, we give a unified proof of both reduced and maximal versions of Mansfield’s Imprimitivity Theorem and obtain a general version of Landstad’s Duality Theorem for twisted group coactions. As one application, we obtain the stabilization trick for arbitrary twisted coactions, showing that every twisted coaction is Morita equivalent to an inflated coaction.
- Alcides Buss and Siegfried Echterhoff, Universal and exotic generalized fixed-point algebras for weakly proper actions and duality, Indiana Univ. Math. J. 63 (2014), no. 6, 1659–1701., DOI 10.1512/iumj.2014.63.5405
- Alcides Buss and Siegfried Echterhoff, Imprimitivity theorems for weakly proper actions of locally compact groups, to appear in Ergodic Theory Dynam. Systems. arXiv:1305.5100.
- Jérôme Chabert and Siegfried Echterhoff, Twisted equivariant $KK$-theory and the Baum-Connes conjecture for group extensions, $K$-Theory 23 (2001), no. 2, 157–200. MR 1857079, DOI 10.1023/A:1017916521415
- Klaus Deicke, Pointwise unitary coactions on $C^*$-algebras with continuous trace, J. Operator Theory 43 (2000), no. 2, 295–327. MR 1753413
- Siegfried Echterhoff, Morita equivalent twisted actions and a new version of the Packer-Raeburn stabilization trick, J. London Math. Soc. (2) 50 (1994), no. 1, 170–186. MR 1277761, DOI 10.1112/jlms/50.1.170
- Siegfried Echterhoff, S. Kaliszewski, and Iain Raeburn, Crossed products by dual coactions of groups and homogeneous spaces, J. Operator Theory 39 (1998), no. 1, 151–176. MR 1610318
- Siegfried Echterhoff, S. Kaliszewski, and John Quigg, Maximal coactions, Internat. J. Math. 15 (2004), no. 1, 47–61. MR 2039211, DOI 10.1142/S0129167X04002107
- Siegfried Echterhoff, S. Kaliszewski, John Quigg, and Iain Raeburn, A categorical approach to imprimitivity theorems for $C^*$-dynamical systems, Mem. Amer. Math. Soc. 180 (2006), no. 850, viii+169. MR 2203930, DOI 10.1090/memo/0850
- Siegfried Echterhoff and Iain Raeburn, The stabilisation trick for coactions, J. Reine Angew. Math. 470 (1996), 181–215. MR 1370212
- Ruy Exel, Unconditional integrability for dual actions, Bol. Soc. Brasil. Mat. (N.S.) 30 (1999), no. 1, 99–124. MR 1686980, DOI 10.1007/BF01235677
- Philip Green, The local structure of twisted covariance algebras, Acta Math. 140 (1978), no. 3-4, 191–250. MR 493349, DOI 10.1007/BF02392308
- Astrid An Huef and Iain Raeburn, Mansfield’s imprimitivity theorem for arbitrary closed subgroups, Proc. Amer. Math. Soc. 132 (2004), no. 4, 1153–1162. MR 2045432, DOI 10.1090/S0002-9939-03-07189-2
- Astrid an Huef, S. Kaliszewski, Iain Raeburn, and Dana P. Williams, Naturality of Rieffel’s Morita equivalence for proper actions, Algebr. Represent. Theory 14 (2011), no. 3, 515–543. MR 2785921, DOI 10.1007/s10468-009-9201-2
- Astrid an Huef, S. Kaliszewski, Iain Raeburn, and Dana P. Williams, Fixed-point algebras for proper actions and crossed products by homogeneous spaces, Illinois J. Math. 55 (2011), no. 1, 205–236 (2012). MR 3006686
- S. Kaliszewski, Magnus B. Landstad, and John Quigg, Exotic group $C^*$-algebras in noncommutative duality, New York J. Math. 19 (2013), 689–711. MR 3141810
- Steven P. Kaliszewski, Magnus B. Landstad, and John Quigg, Exotic coactions (2013), preprint. arXiv:1305.5489.
- S. Kaliszewski and John Quigg, Imprimitivity for $C^\ast$-coactions of non-amenable groups, Math. Proc. Cambridge Philos. Soc. 123 (1998), no. 1, 101–118. MR 1474869, DOI 10.1017/S0305004197001692
- S. Kaliszewski and John Quigg, Mansfield’s imprimitivity theorem for full crossed products, Trans. Amer. Math. Soc. 357 (2005), no. 5, 2021–2042. MR 2115089, DOI 10.1090/S0002-9947-04-03683-9
- S. Kaliszewski, John Quigg, and Iain Raeburn, Proper actions, fixed-point algebras and naturality in nonabelian duality, J. Funct. Anal. 254 (2008), no. 12, 2949–2968. MR 2418615, DOI 10.1016/j.jfa.2008.03.010
- Yoshikazu Katayama, Takesaki’s duality for a nondegenerate co-action, Math. Scand. 55 (1984), no. 1, 141–151. MR 769030, DOI 10.7146/math.scand.a-12072
- Eberhard Kirchberg and Simon Wassermann, Permanence properties of $C^*$-exact groups, Doc. Math. 4 (1999), 513–558. MR 1725812
- Kevin Mansfield, Induced representations of crossed products by coactions, J. Funct. Anal. 97 (1991), no. 1, 112–161. MR 1105657, DOI 10.1016/0022-1236(91)90018-Z
- John Phillips and Iain Raeburn, Twisted crossed products by coactions, J. Austral. Math. Soc. Ser. A 56 (1994), no. 3, 320–344. MR 1271525, DOI 10.1017/S1446788700035539
- John C. Quigg, Landstad duality for $C^*$-coactions, Math. Scand. 71 (1992), no. 2, 277–294. MR 1212711, DOI 10.7146/math.scand.a-12429
- John C. Quigg, Full and reduced $C^*$-coactions, Math. Proc. Cambridge Philos. Soc. 116 (1994), no. 3, 435–450. MR 1291751, DOI 10.1017/S0305004100072728
- John C. Quigg and Iain Raeburn, Induced $C^*$-algebras and Landstad duality for twisted coactions, Trans. Amer. Math. Soc. 347 (1995), no. 8, 2885–2915. MR 1297536, DOI 10.1090/S0002-9947-1995-1297536-3
- Marc A. Rieffel, Proper actions of groups on $C^*$-algebras, Mappings of operator algebras (Philadelphia, PA, 1988) Progr. Math., vol. 84, Birkhäuser Boston, Boston, MA, 1990, pp. 141–182. MR 1103376
- Marc A. Rieffel, Integrable and proper actions on $C^*$-algebras, and square-integrable representations of groups, Expo. Math. 22 (2004), no. 1, 1–53. MR 2166968, DOI 10.1016/S0723-0869(04)80002-1
- Dana P. Williams, Crossed products of $C{^\ast }$-algebras, Mathematical Surveys and Monographs, vol. 134, American Mathematical Society, Providence, RI, 2007. MR 2288954, DOI 10.1090/surv/134
Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 46L55, 22D35
Retrieve articles in all journals with MSC (2010): 46L55, 22D35
Additional Information
Alcides Buss
Affiliation:
Departamento de Matemática, Universidade Federal de Santa Catarina, 88.040-900 Florianópolis-SC, Brazil
MR Author ID:
827256
ORCID:
0000-0001-6796-9818
Email:
alcides.buss@ufsc.br
Siegfried Echterhoff
Affiliation:
Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, 48149 Münster, Germany
MR Author ID:
266728
ORCID:
0000-0001-9443-6451
Email:
echters@uni-muenster.de
Keywords:
Weakly proper group action,
generalized fixed-point algebra,
Mansfield Imprimitivity Theorem,
exotic crossed product,
twisted group coactions,
Landstad Duality
Received by editor(s):
October 30, 2013
Published electronically:
March 4, 2015
Additional Notes:
This research was supported by Deutsche Forschungsgemeinschaft (SFB 878, Groups, Geometry & Actions) and by CNPq (Ciências sem Fronteira) – Brazil.
Article copyright:
© Copyright 2015
American Mathematical Society
