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Extension problems and non-Abelian duality for C*-algebras

  • Astrid an Huef (a1), S. Kaliszewski (a2) and Iain Raeburn (a3)

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Suppose that H is a closed subgroup of a locally compact group G. We show that a unitary representation U of H is the restriction of a unitary representation of G if and only if a dual representation Û of a crossed product C*(G) ⋊ (G/H) is regular in an appropriate sense. We then discuss the problem of deciding whether a given representation is regular; we believe that this problem will prove to be an interesting test question in non-Abelian duality for crossed products of C*-algebras.

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References

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[1]Deicke, K., Pask, D. and Raeburn, I., ‘Coverings of directed graphs and crossed products of C*-algebras by coactions of homogeneous spaces’, Internat. J. Math. 14 (2003), 773789.
[2]Echterhoff, S., ‘Duality of induction and restriction for abelian twisted covariant systems’, Math. Proc. Camb. Phil. Soc. 116 (1994), 301315.
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[4]Echterhoff, S., Kaliszewski, S., Quigg, J. and Raeburn, I., ‘A categorical approach to imprimitivity theorems for C*-dynamical systems’, Memoirs Amer. Math. Soc. 180 (2006), 1169.
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[7]Green, P., ‘The local structure of twisted covariance algebras’, Acta Math. 140 (1978), 191250.
[8]an Huef, A., Kaliszewski, S. and Raeburn, I., ‘Covariant representations of Hecke algebras and imprimitivity for crossed products by homogeneous spaces’, (preprint) arXiv:math.0A/0509291.
[9]an Huef, A. and Raeburn, I., ‘Twisted actions and the obstruction to extending unitary representations of subgroups’, J. Pure Appl. Algebra 194 (2004), 299309.
[10]Kaliszewski, S. and Quigg, J., ‘Mansfield imprimitivity for full crossed products’, Trans. Amer. Math. Soc. 357 (2005), 20212042.
[11]Kaliszewski, S., Quigg, J. and Raeburn, I., ‘Duality of restriction and induction for C*-coactions’, Trans. Amer. Math. Soc. 349 (1997), 20852113.
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[13]Pask, D., Quigg, J. and Raeburn, I., ‘Coverings of k-graphs’, J. Algebra 289 (2005), 161191.
[14]Raeburn, I. and Williams, D.P., Morita Equivalence and Continuous-Trace C*-Algebras, Math. Surveys and Monographs 60 (Amer. Math. Soc, Providence, R.I., 1998).
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Extension problems and non-Abelian duality for C*-algebras

  • Astrid an Huef (a1), S. Kaliszewski (a2) and Iain Raeburn (a3)

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