Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-05T10:50:40.244Z Has data issue: false hasContentIssue false
  • English
  • Français

Three bimodules for Mansfield's imprimitivity theorem

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  09 April 2009

S. Kaliszewski  and
John Quigg
S. Kaliszewski
Affiliation:
Department of Mathematics, Arizona State University, Tempe AZ 85287, USA e-mail: [email protected] e-mail: [email protected]
John Quigg
Affiliation:
Department of Mathematics, Arizona State University, Tempe AZ 85287, USA e-mail: [email protected] e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For a maximal coaction δ of a discrete group G on a C*-algebra A and a normal subgroup N of G, there are at least three natural A × G ×δ| N - A ×δ|1 G/N imprimitivity bimodules: Mansfield's bimodule ; the bimodule assembled by Ng from Green's imprimitivity bimodule and Katayama duality; and the bimodule assembled from and the crossed-product Mansfield bimodule . We show that all three of these are isomorphic, so that the corresponding inducing maps on representations are identical. This can be interpreted as saying that Mansfield and Green induction are inverses of one another ‘modulo Katayama duality’. These results pass to twisted coactions; dual results starting with an action are also given.

Keywords

C*-algebracoactionduality

MSC classification

Secondary: 46L55: Noncommutative dynamical systems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 71 , Issue 3 , December 2001 , pp. 397 - 420
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Echterhoff, S., Kaliszewski, S. and Quigg, J., ‘Maximal coactions’, preprint, 2000.Google Scholar
[2]Echterhoff, S., Kaliszewski, S., Quigg, J. and Raeburn, I., ‘Naturality and induced representations’, Bull. Austral. Math. Soc. 61 (2000), 415–438.CrossRefGoogle Scholar
[3]Echterhoff, S. and Quigg, J., ‘Full duality for coactions of discrete groups’, Math. Scand., to appear.Google Scholar
[4]Echterhoff, S. and Quigg, J., ‘Induced coactions of discrete groups on C*-algebras’, Canad. J. Math. 51 (1999), 745–770.CrossRefGoogle Scholar
[5]Echterhoff, S. and Raeburn, I., ‘Multipliers of imprimitivity bimodules and Morita equivalence of crossed products’, Math. Scand. 76 (1995), 289–309.Google Scholar
[6]Green, P., ‘The local structure of twisted covanance algebras’, Acta Math. 140 (1978), 191–250.Google Scholar
[7]Kaliszewski, S. and Quigg, J., ‘Imprimitivity for C*-coactions of non-amenable groups’, Math. Proc. Cambridge Philos. Soc. 123 (1998), 101–118.Google Scholar
[8]Kaliszewski, S., Quigg, J. and Raeburn, I., ‘Duality of restriction and induction for C*-coactions’, Trans. Amer. Math. Soc. 349 (1997), 2085–2113.Google Scholar
[9]Katayama, Y., ‘Takesaki's duality for a non-degenerate co-action’, Math. Scand. 55 (1985), 141–151.CrossRefGoogle Scholar
[10]Landstad, M. B., Phillips, J., Raeburn, I. and Sutherland, C. E., ‘Representations of crossed products by coactions and principal bundles’, Trans. Amer Math. Soc. 299 (1987), 747–784.Google Scholar
[11]Mansfield, K., ‘Induced representations of crossed products by coactions’, J. Funct. Anal. 97 (1991), 112–161.Google Scholar
[12]Ng, C. K., ‘A remark on Mansfield's imprimitivity theorem’, Proc. Amer. Math. Soc. 126 (1998), 3767–3768.Google Scholar
[13]Phillips, J. and Raeburn, I., ‘Twisted crossed products by coactions’, J. Austral. Math. Soc. Ser A 56 (1994), 320–344.Google Scholar
[14]Quigg, J., ‘Full and reduced C*-coactions, Math. Proc. Cambridge Philos. Soc. 116 (1994), 435–450.CrossRefGoogle Scholar
[15]Quigg, J. and Raeburn, I., ‘Induced C*-algebras and Landstad duality for twisted coactions’, Trans. Amer Math. Soc. 347 (1995), 2885–2915.Google Scholar
[16]Raeburn, I., ‘On crossed products by coactions and their representation theory’, Proc. London Math. Soc. 64 (1992), 625–652.Google Scholar
[17]Sieben, N., ‘Morita equivalence of C*-crossed products by inverse semigroup actions’, Rocky Mountain J. Math., to appear.Google Scholar