Mansfield’s imprimitivity theorem for arbitrary closed subgroups

an Huef, Astrid; Raeburn, Iain

doi:10.1090/S0002-9939-03-07189-2

Mansfield’s imprimitivity theorem for arbitrary closed subgroups
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Proc. Amer. Math. Soc. 132 (2004), 1153-1162 Request permission

Abstract:

Let $\delta$ be a nondegenerate coaction of $G$ on a $C^*$-algebra $B$, and let $H$ be a closed subgroup of $G$. The dual action $\hat \delta :H\to \operatorname {Aut}(B\times _\delta G)$ is proper and saturated in the sense of Rieffel, and the generalised fixed-point algebra is the crossed product of $B$ by the homogeneous space $G/H$. The resulting Morita equivalence is a version of Mansfield’s imprimitivity theorem which requires neither amenability nor normality of $H$.

References

K. Deicke, D. Pask, and I. Raeburn, Coverings of directed graphs and crossed products of $C^*$-algebras by coactions of homogeneous spaces, Internat. J. Math., to appear.
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 and John Quigg, Full duality for coactions of discrete groups, Math. Scand. 90 (2002), no. 2, 267–288. MR 1895615, DOI 10.7146/math.scand.a-14374
Philip Green, The local structure of twisted covariance algebras, Acta Math. 140 (1978), no. 3-4, 191–250. MR 493349, DOI 10.1007/BF02392308
A. an Huef, I. Raeburn, and D. P. Williams, A symmetric imprimitivity theorem for commuting proper actions, preprint (arXiv.math.OA/0202046), 2002.
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
M. B. Landstad, J. Phillips, I. Raeburn, and C. E. Sutherland, Representations of crossed products by coactions and principal bundles, Trans. Amer. Math. Soc. 299 (1987), no. 2, 747–784. MR 869232, DOI 10.1090/S0002-9947-1987-0869232-0
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
Chi-Keung Ng, A remark on Mansfield’s imprimitivity theorem, Proc. Amer. Math. Soc. 126 (1998), no. 12, 3767–3768. MR 1626462, DOI 10.1090/S0002-9939-98-05066-7
Dorte Olesen and Gert K. Pedersen, Applications of the Connes spectrum to $C^{\ast }$-dynamical systems, J. Functional Analysis 30 (1978), no. 2, 179–197. MR 515224, DOI 10.1016/0022-1236(78)90069-1
Dorte Olesen and Gert K. Pedersen, Applications of the Connes spectrum to $C^{\ast }$-dynamical systems, J. Functional Analysis 30 (1978), no. 2, 179–197. MR 515224, DOI 10.1016/0022-1236(78)90069-1
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
Iain Raeburn, On crossed products by coactions and their representation theory, Proc. London Math. Soc. (3) 64 (1992), no. 3, 625–652. MR 1153000, DOI 10.1112/plms/s3-64.3.625
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
M. A. Rieffel, Integrable and proper actions on $C^*$-algebras, and square integrable representations of groups, preprint (arXiv.math.OA/9809098), 1999.