Duality for full crossed products of 𝐶*-algebras by non-amenable groups

Nilsen, May

doi:10.1090/S0002-9939-98-04598-5

Duality for full crossed products of $C^{\displaystyle *}\!$ -algebras by non-amenable groups

Author: May Nilsen
Journal: Proc. Amer. Math. Soc. 126 (1998), 2969-2978
MSC (1991): Primary 46L55
DOI: https://doi.org/10.1090/S0002-9939-98-04598-5
MathSciNet review: 1469427
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Abstract: Let $(A, G, \delta)$ be a cosystem and $(A, G,\alpha)$ be a dynamical system. We examine the extent to which induction and restriction of ideals commute, generalizing some of the results of Gootman and Lazar (1989) to full crossed products by non-amenable groups. We obtain short, new proofs of Katayama and Imai-Takai duality, the faithfulness of the induced regular representation for full coactions and actions by amenable groups. We also give a short proof that the space of dual-invariant ideals in the crossed product is homeomorphic to the space of invariant ideals in the algebra, and give conditions under which the restriction mapping is open.

1. E. C. Gootman and A. J. Lazar, Applications of non-commutative duality to crossed product $C^{\displaystyle *}\!$ -algebras determined by an action or coaction, Proc. London Math. Soc. 59 (1989), 593-624. MR 91b:46069
2. P. Green, The local structure of twisted covariance algebras, Acta. Math. 140 (1978), 191-250. MR 58:12376
3. S. Imai and H. Takai, On a duality for $C^{\displaystyle *}\!$ -crossed products by a locally compact group, J. Math. Soc. Japan 30 (1978), 495-504. MR 81h:46090
4. S. Kaliszewski, J. Quigg, and I. Raeburn, Duality of restriction and induction for $C^{\displaystyle *}\!$ -coactions, Trans. Amer. Math. Soc. 349 (1997), 2085-2113. CMP 97:09
5. Y. Katayama, Takesaki's duality for a non-degenerate co-action, Math. Scand. 55 (1984), 141-151. MR 86b:46112
6. K. Mansfield, Induced representations of crossed products by coactions, J. Funct. Anal. 97 (1991), 112-161. MR 92h:46095
7. Y. Nakagami and M. Takesaki, Duality for crossed products of von Neumann algebras, Springer-Verlag, Berlin, 1979. MR 81e:46053
8. M. Nilsen, Full crossed products by coactions, C ${}_0$ (X)-algebras and $C^{\displaystyle *}\!$ -bundles, to appear, Bull. London Math. Soc.
9. -, $C^{\displaystyle *}\!$ -bundles and C ${}_0$ (X)-algebras, Indiana Univ. Math. J. 45 (1996), 463-477. CMP 97:02
10. J. C. Quigg, Full and reduced $C^{\displaystyle *}\!$ -coactions, Math. Proc. Camb. Philos. Soc. 116 (1994), 435-450. MR 95g:46126
11. I. Raeburn, On crossed products and Takai duality, Proc. Edin. Math. Soc. 31 (1988), 321-330. MR 90d:46093
12. -, On crossed products by coactions and their representation theory, Proc. London Math. Soc. 64 (1992), 625-652. MR 93e:46080