Abstract
An exotic crossed product is a way of associating a C ∗-algebra to each C ∗-dynamical system that generalizes the well-known universal and reduced crossed products. Exotic crossed products provide natural generalizations of, and tools to study, exotic group C ∗-algebras as recently considered by Brown-Guentner and others. They also form an essential part of a recent program to reformulate the Baum-Connes conjecture with coefficients so as to mollify the counterexamples caused by failures of exactness.In this paper, we survey some constructions of exotic group algebras and exotic crossed products. Summarising our earlier work, we single out a large class of crossed products—the correspondence functors—that have many properties known for the maximal and reduced crossed products: for example, they extend to categories of equivariant correspondences, and have a compatible descent morphism in K K-theory. Combined with known results on K-amenability and the Baum-Connes conjecture, this allows us to compute the K-theory of many exotic group algebras. It also gives new information about the reformulation of the Baum-Connes Conjecture mentioned above. Finally, we present some new results relating exotic crossed products for a group and its closed subgroups, and discuss connections with the reformulated Baum-Connes conjecture.
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Notes
- 1.
We always assume inner products on Hilbert spaces to be linear in the second variable.
- 2.
Added in proof: these ingredients recently appeared for second countable G in a paper of Brodzki, Cave, and Li, arXiv.1603.01829.
References
R.J. Archbold, and J.S. Spielberg, Topologically free actions and ideals in discrete C ∗ -dynamical systems, Proc. Edinburgh Math. Soc. (2), 37 (1994), 119–124.
P. Baum, A. Connes, and N. Higson. Classifying space for proper actions and K -theory of group C ∗ -algebras, in C ∗-Algebras: 1943–1993, San Antonio, TX, 1993, Contemp. Math. 167. Amer. Math. Soc., Providence, RI 1994. p. 240–291.
P. Baum, E. Guentner, and R. Willett, Expanders, exact crossed products, and the Baum-Connes conjecture, Ann. K-theory 1 (2015), 155–208.
N. P. Brown and E. Guentner, New C ∗ -completions of discrete groups and related spaces, Bull. Lond. Math. Soc., 45 (2013), 1181–1193.
N. P. Brown and N. Ozawa, C ∗-algebras and finite-dimensional approximations, Graduate Studies in Mathematics 88, American Mathematical Society, Providence, RI, 2008, pages xvi+509.
A. Buss and S. Echterhoff, Universal and exotic generalized fixed-point algebras for weakly proper actions and duality, Indiana Univ. Math. J. 63 (2014), 1659–1701.
A. Buss and S. Echterhoff, Imprimitivity theorems for weakly proper actions of locally compact groups, Ergodic Theory Dynam. Systems 35 (2015), 2412–2457, doi 10.1017/etds.2014.36.
A. Buss and S. Echterhoff, Maximality of dual coactions on sectional C ∗ -algebras of Fell bundles and applications, Studia Math. 229 (2015), 233–262, doi 10.4064/sm8361-1-2016.
A. Buss, S. Echterhoff, and R. Willett, Exotic crossed products and the Baum-Connes conjecture, J. Reine Angew. Math. doi 10.1515/crelle-2015-0061 (prepublished electronically).
J. Chabert and S. Echterhoff, Permanence properties of the Baum-Connes conjecture, Doc. Math., 6 (2001), 127–183 (electronic).
J. Cuntz, K -theoretic amenability for discrete groups, J. Reine Angew. Math. 344 (1983), 180–195.
S. Echterhoff, Crossed products, the Mackey-Rieffel-Green machine and applications, eprint, arxiv 1006.4975v2, 2010.
S. Echterhoff, Morita equivalent twisted actions and a new version of the Packer–Raeburn stabilization trick, J. London Math. Soc. (2) 50 (1994), 170–186.
S. Echterhoff, S. P. Kaliszewski, J. Quigg, and I. Raeburn, A categorical approach to imprimitivity theorems for C ∗ -dynamical systems, Mem. Amer. Math. Soc. 180 (2006) no. 850, pp. viii+169.
P. Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France, 92 (1964), 181–236.
P. Green, Philip, The local structure of twisted covariance algebras, Acta Math. 140 (1978), 191–250.
M. Gromov, Random walks in random groups, Geom. Funct. Anal. 13 (2003), 73–146.
U. Haagerup, An example of a non nuclear C ∗ -algebra that has the metric approximation property, Invent. Math. 50 (1979), 289–293.
R. Howe and E.C. Tan, Non-abelian harmonic analysis, Universitext, Springer-Verlag, New York, NY, 1992, pp. xv+357.
N. Higson, V. Lafforgue, and G. Skandalis, Counterexamples to the Baum-Connes conjecture, Geom. Funct. Anal. 12 (2002), 330–354.
A. Hulanicki, Means and Følner conditions on locally compact groups, Studia Math. 27 (1966), 87–104.
P. Jolissaint, Notes on C0-representations and the Haagerup property, Bull. Belg. Math. Soc. Simon Stevin 21 (2014), 263–274.
P. Julg and A. Valette, K -theoretic amenability for SL2(Q p ), and the action on the associated tree, J. Funct. Anal., 58 (1984), 194–215.
S. P. Kaliszewski, M. B. Landstad, and J. Quigg, Exotic group C ∗ -algebras in noncommutative duality, New York J. Math. 19 (2013), 689–711.
S. P. Kaliszewski, M. B. Landstad, and J. Quigg, Exotic coactions, preprint 2013, arxiv1305.5489.
S. P. Kaliszewski, M. B. Landstad, and J. Quigg, Coaction functors, preprint 2015, arxiv1505.03487.
D. Kerr, C ∗ -algebras and topological dynamics: finite approximations and paradoxicality, 2011, available on the author’s website.
E. Kirchberg and S. Wassermann, Exact groups and continuous bundles of C ∗ -algebras, Math. Ann. 315 (1999), 169–203.
E. Kirchberg and S. Wassermann, Permanence properties of C*-exact groups, Doc. Math. 4 (1999), 513–558 (electronic).
R. Kunze and E. Stein, Uniformly bounded representations and harmonic analysis of the2 × 2real unimodular group, Amer. J. Math. 82 (1960), 1–62.
D. Miličić, Topological representation of the group C ∗ -algebra of \(SL(2, \mathbb{R})\), Glasnik Mat. Ser. III 6(26) (1971), 231–246.
R. Okayasu, Free group C*-algebras associated with ℓ p , Internat. J. Math. 25 (2014), 1450065 (12 pages).
D. Osajda, Small cancellation labellings of some infinite graphs and applications, eprint 2014, arxiv 1406.5015.
R. Scaramuzzi, A notion of rank for unitary representations of general linear groups, Trans. Amer. Math. Soc. 319 (1990), 349–379.
D. P. Williams, Crossed products of C ∗-algebras, Mathematical Surveys and Monographs 134, American Mathematical Society, Providence, RI, 2007, pp. xvi+528.
M. Wiersma, L p -Fourier and Fourier-Stieltjes algebras for locally compact groups, J. Funct. Anal. 269 (2015), 3928–3951.
Acknowledgements
Part of the work on this paper took place during a visit of the first and third authors to the Westfälische Wilhelms-Universität, Münster. We would like to thank this institution for its hospitality.
This work has been supported by Deutsche Forschungsgemeinschaft (SFB 878, Groups, Geometry & Actions), by CNPq/CAPES—Brazil, and by the US NSF (DMS 1401126).
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Buss, A., Echterhoff, S., Willett, R. (2016). Exotic Crossed Products. In: Carlsen, T.M., Larsen, N.S., Neshveyev, S., Skau, C. (eds) Operator Algebras and Applications. Abel Symposia, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-39286-8_3
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