Weakly proper group actions, Mansfield’s Imprimitivity and twisted Landstad duality

Buss, Alcides; Echterhoff, Siegfried

doi:10.1090/S0002-9947-2015-06406-X

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

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.

[1] 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.
[2] 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.
[3] 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, https://doi.org/10.1023/A:1017916521415
[4] Klaus Deicke, Pointwise unitary coactions on 𝐶*-algebras with continuous trace, J. Operator Theory 43 (2000), no. 2, 295–327. MR 1753413
[5] 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, https://doi.org/10.1112/jlms/50.1.170
[6] 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
[7] Siegfried Echterhoff, S. Kaliszewski, and John Quigg, Maximal coactions, Internat. J. Math. 15 (2004), no. 1, 47–61. MR 2039211, https://doi.org/10.1142/S0129167X04002107
[8] 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, https://doi.org/10.1090/memo/0850
[9] Siegfried Echterhoff and Iain Raeburn, The stabilisation trick for coactions, J. Reine Angew. Math. 470 (1996), 181–215. MR 1370212
[10] Ruy Exel, Unconditional integrability for dual actions, Bol. Soc. Brasil. Mat. (N.S.) 30 (1999), no. 1, 99–124. MR 1686980, https://doi.org/10.1007/BF01235677
[11] Philip Green, The local structure of twisted covariance algebras, Acta Math. 140 (1978), no. 3-4, 191–250. MR 493349, https://doi.org/10.1007/BF02392308
[12] 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, https://doi.org/10.1090/S0002-9939-03-07189-2
[13] 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, https://doi.org/10.1007/s10468-009-9201-2
[14] 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
[15] 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
[16] Steven P. Kaliszewski, Magnus B. Landstad, and John Quigg, Exotic coactions (2013), preprint. arXiv:1305.5489.
[17] S. Kaliszewski and John Quigg, Imprimitivity for 𝐶*-coactions of non-amenable groups, Math. Proc. Cambridge Philos. Soc. 123 (1998), no. 1, 101–118. MR 1474869, https://doi.org/10.1017/S0305004197001692
[18] 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, https://doi.org/10.1090/S0002-9947-04-03683-9
[19] 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, https://doi.org/10.1016/j.jfa.2008.03.010
[20] Yoshikazu Katayama, Takesaki’s duality for a nondegenerate co-action, Math. Scand. 55 (1984), no. 1, 141–151. MR 769030, https://doi.org/10.7146/math.scand.a-12072
[21] Eberhard Kirchberg and Simon Wassermann, Permanence properties of 𝐶*-exact groups, Doc. Math. 4 (1999), 513–558. MR 1725812
[22] Kevin Mansfield, Induced representations of crossed products by coactions, J. Funct. Anal. 97 (1991), no. 1, 112–161. MR 1105657, https://doi.org/10.1016/0022-1236(91)90018-Z
[23] John Phillips and Iain Raeburn, Twisted crossed products by coactions, J. Austral. Math. Soc. Ser. A 56 (1994), no. 3, 320–344. MR 1271525
[24] John C. Quigg, Landstad duality for 𝐶*-coactions, Math. Scand. 71 (1992), no. 2, 277–294. MR 1212711, https://doi.org/10.7146/math.scand.a-12429
[25] John C. Quigg, Full and reduced 𝐶*-coactions, Math. Proc. Cambridge Philos. Soc. 116 (1994), no. 3, 435–450. MR 1291751, https://doi.org/10.1017/S0305004100072728
[26] 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, https://doi.org/10.1090/S0002-9947-1995-1297536-3
[27] 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
[28] 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, https://doi.org/10.1016/S0723-0869(04)80002-1
[29] Dana P. Williams, Crossed products of 𝐶*-algebras, Mathematical Surveys and Monographs, vol. 134, American Mathematical Society, Providence, RI, 2007. MR 2288954