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Exotic Coactions

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  09 March 2016

S. Kaliszewski ,
Magnus B. Landstad  and
John Quigg
S. Kaliszewski
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA (kaliszewski@asu.edu; quigg@asu.edu)
Magnus B. Landstad
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway (magnusla@math.ntnu.no)
John Quigg
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA (kaliszewski@asu.edu; quigg@asu.edu)
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Abstract

If a locally compact group G acts on a C*-algebra B, we have both full and reduced crossed products and each has a coaction of G. We investigate ‘exotic’ coactions in between the two, which are determined by certain ideals E of the Fourier–Stieltjes algebra B(G); an approach that is inspired by recent work of Brown and Guentner on new C*-group algebra completions. We actually carry out the bulk of our investigation in the general context of coactions on a C*-algebra A. Buss and Echterhoff have shown that not every coaction comes from one of these ideals, but nevertheless the ideals do generate a wide array of exotic coactions. Coactions determined by these ideals E satisfy a certain ‘E-crossed product duality’, intermediate between full and reduced duality. We give partial results concerning exotic coactions with the ultimate goal being a classification of which coactions are determined by ideals of B(G).

Keywords

group C*-algebracoactionC*-bialgebraFourier–Stieltjes algebra

MSC classification

Primary: 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 2 , May 2016 , pp. 411 - 434
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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References

1. An Huef, A., Quigg, J., Raeburn, I. and Williams, D. P., Full and reduced coactions of locally compact groups on C *-algebras, Exp. Math. 29 (2011), 3–23.Google Scholar
2. Brown, N. P. and Guentner, E., New C * -completions of discrete groups and related spaces, Bull. Lond. Math. Soc. 45 (2013), 1181–1193.Google Scholar
3. Brown, N. P. and Ozawa, N., C * -algebras and finite-dimensional approximations, Graduate Studies in Mathematics, Volume 88 (American Mathematical Society, Providence, RI, 2008).Google Scholar
4. Buss, A. and Echterfhoff, S., Imprimitivity theorems for weakly proper actions of locally compact groups, Ergod. Theory Dynam. Syst. 35 (2015), 2412–2457.Google Scholar
5. Buss, A. and Echterfhoff, S., Universal and exotic generalized fixed-point algebras for weakly proper actions and duality, Indiana Univ. Math. J. 63 (2014), 1659–1701.CrossRefGoogle Scholar
6. Echterhoff, S., Kaliszewski, S. and Quigg, J., Maximal coactions, Int. J. Math. 15 (2004), 47–61.Google Scholar
7. Echterhoff, S., Kaliszewski, S., Quigg, J. and Raeburn, I., A categorical approach to imprimitivity theorems for C *-dynamical systems, Memoirs of the American Mathematical Society, Volume 180, Number 850 (American Mathematical Society, Providence, RI, 2006).Google Scholar
8. Ellwood, D., A new characterisation of principal actions, J. Funct. Analysis 173(1) (2000), 49–60.CrossRefGoogle Scholar
9. Goswami, D. and Kuku, A. O., A complete formulation of the Baum–Connes conjecture for the action of discrete quantum groups, K-Theory 30(4) (2003), 341–363.Google Scholar
10. Kaliszewski, S. and Quigg, J., Imprimitivity for C *-coactions of non-amenable groups, Math. Proc. Camb. Phil. Soc. 123 (1998), 101–118.Google Scholar
11. Kaliszewski, S., Landstad, M. B. and Quigg, J., Exotic group C *-algebras in noncommutative duality, New York J. Math. 19 (2013), 689–711.Google Scholar
12. Landstad, M. B., Phillips, J., Raeburn, I. and Sutherland, C. E., Representations of crossed products by coactions and principal bundles, Trans. Am. Math. Soc. 299 (1987), 747–784.Google Scholar
13. Okayasu, R., Free group C *-algebras associated with lp , Preprint (arXiv:1203.0800; 2012).Google Scholar
14. Quigg, J. C., Full and reduced C *-coactions, Math. Proc. Camb. Phil. Soc. 116 (1994), 435–450.Google Scholar