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The maximal injective crossed product

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  22 April 2019

ALCIDES BUSS ,
SIEGFRIED ECHTERHOFF  and
RUFUS WILLETT
ALCIDES BUSS
Affiliation:
Departamento de Matemática, Universidade Federal de Santa Catarina, 88.040-900Florianópolis-SC, Brazil email [email protected]
SIEGFRIED ECHTERHOFF
Affiliation:
Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, 48149Münster, Germany email [email protected]
RUFUS WILLETT
Affiliation:
Mathematics Department, University of Hawai‘i at Mānoa, Keller 401A, 2565 McCarthy Mall, Honolulu, HI96822, USA email [email protected]
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Abstract

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A crossed product functor is said to be injective if it takes injective morphisms to injective morphisms. In this paper we show that every locally compact group $G$ admits a maximal injective crossed product $A\mapsto A\rtimes _{\text{inj}}G$. Moreover, we give an explicit construction of this functor that depends only on the maximal crossed product and the existence of $G$-injective $C^{\ast }$-algebras; this is a sort of ‘dual’ result to the construction of the minimal exact crossed product functor, the latter having been studied for its relationship to the Baum–Connes conjecture. It turns out that $\rtimes _{\text{inj}}$ has interesting connections to exactness, the local lifting property, amenable traces, and the weak expectation property.

Keywords

exotic crossed productsinjective C∗ -algebrasexact groupslocal lifting propertyweak expectation property

MSC classification

Primary: 46L55: Noncommutative dynamical systems
Secondary: 46L80: $K$-theory and operator algebras (including cyclic theory)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 11 , November 2020 , pp. 2995 - 3014
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

References

Anantharaman-Delaroche, C.. Amenability and exactness for dynamical systems and their C ∗ -algebras. Trans. Amer. Math. Soc. 354(10) (2002), 4153–4178.CrossRefGoogle Scholar
Arhangel’skii, A.. A study of extremally disconnected topological spaces. Bull. Math. Sci. 1(1) (2011), 3–12.CrossRefGoogle Scholar
Baum, P., Guentner, E. and Willett, R.. Expanders, exact crossed products, and the Baum–Connes conjecture. Ann. K-theory 1(2) (2015), 155–208.CrossRefGoogle Scholar
Bhattacharya, A. and Farenick, D.. Crossed products of C ∗ -algebras with the weak expectation property. New York J. Math. 19 (2013), 423–429.Google Scholar
Blackadar, B.. Operator Algebras: Theory of C ∗ -Algebras and Von Neumann Algebras. Springer, Berlin, 2006.CrossRefGoogle Scholar
Brodzki, J., Cave, C. and Li, K.. Exactness of locally compact second countable groups. Adv. Math. 312 (2017), 209–233.CrossRefGoogle Scholar
Brown, N.. Invariant means and finite representation theory of C ∗ -algebras. Mem. Amer. Math. Soc. 184(865) (2006), vi+105pp.Google Scholar
Brown, N. and Guentner, E.. New C ∗ -completions of discrete groups and related spaces. Bull. Lond. Math. Soc. 45(6) (2013), 1181–1193.CrossRefGoogle Scholar
Brown, N. and Ozawa, N.. C ∗ -Algebras and Finite-Dimensional Approximations (Graduate Studies in Mathematics, 88) . American Mathematical Society, Providence, RI, 2008.CrossRefGoogle Scholar
Buss, A. and Echterhoff, S.. Maximality of dual coactions on sectional C ∗ -algebras of Fell bundles and applications. Studia Math. 229(3) (2015), 233–262.Google Scholar
Buss, A., Echterhoff, S. and Willett, R.. Exotic crossed products. Operator Algebras and Applications (The Abel Symposium) . Eds. Carlsen, T. M., Larsen, N. S., Neshveyev, S. and Skau, C.. Springer, Cham, 2015, pp. 61–108.Google Scholar
Buss, A., Echterhoff, S. and Willett, R.. Exotic crossed products and the Baum–Connes conjecture. J. Reine Angew. Math. 740 (2018), 111–159.CrossRefGoogle Scholar
Buss, A., Echterhoff, S. and Willett, R.. The minimal exact crossed product. Documenta Math. 23 (2018), 2043–2077.Google Scholar
Cuntz, J., Echterhoff, S., Li, X. and Yu, G.. K-Theory for Group C ∗ -Algebras and Semigroup C ∗ -Algebras (Oberwolfach Seminars) . Birkhäuser, Cham, 2017.CrossRefGoogle Scholar
Echterhoff, S., Kaliszewski, S., Quigg, J. and Raeburn, I.. A categorical approach to imprimitivity theorems for C ∗ -dynamical systems. Mem. Amer. Math. Soc. 180 (2006), viii+169 pp.Google Scholar
Guentner, E., Tessera, R. and Yu, G.. Discrete groups with finite decomposition complexity. Groups Geom. Dyn. 7(2) (2013), 377–402.CrossRefGoogle Scholar
Hamana, M.. Injective envelopes of C ∗ -dynamical systems. Tohoku Math. J. 37 (1985), 463–487.CrossRefGoogle Scholar
Kaliszewski, S., Landstad, M. and Quigg, J.. Exotic group C ∗ -algebras in noncommutative duality. New York J. Math. 19 (2013), 689–711.Google Scholar
Kaplansky, I.. Projections in Banach algebras. Ann. of Math. (2) 53 (1951), 235–249.CrossRefGoogle Scholar
Osajda, D.. Small cancellation labellings of some infinite graphs and application. Preprint, 2014,arXiv:1406.5015.Google Scholar
Ozawa, N.. Amenable actions and exactness for discrete groups. C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), 691–695.CrossRefGoogle Scholar
Ozawa, N.. About the QWEP conjecture. Internat. J. Math. 15 (2004), 501–530.CrossRefGoogle Scholar
Ozawa, N.. There is no separable universal II 1 -factor. Proc. Amer. Math. Soc. 132(2) (2004), 487–490.CrossRefGoogle Scholar
Roe, J. and Willett, R.. Ghostbusting and property A. J. Funct. Anal. 266(3) (2014), 1674–1684.CrossRefGoogle Scholar
Ruan, Z.-J. and Wiersma, M.. On exotic group C ∗ -algebras. J. Funct. Anal. 271(2) (2016), 437–453.CrossRefGoogle Scholar
Thom, A.. Examples of hyperlinear groups without the factorization property. Groups Geom. Dyn. 4 (2010), 195–208.CrossRefGoogle Scholar
Wiersma, M.. L p -Fourier and Fourier–Stieltjes algebras for locally compact groups. J. Funct. Anal. 269(12) (2015), 3928–3951.CrossRefGoogle Scholar
Wiersma, M.. Constructions of exotic group C ∗ -algebras. Illinois J. Math. 60(3–4) (2016), 655–667.CrossRefGoogle Scholar
Willett, R. and Yu, G.. Higher index theory for certain expanders and Gromov monster groups I. Adv. Math. 229(3) (2012), 1380–1416.CrossRefGoogle Scholar
Willett, R. and Yu, G.. Higher index theory for certain expanders and Gromov monster groups II. Adv. Math. 229(3) (2012), 1762–1803.CrossRefGoogle Scholar
Willett, R. and Yu, G.. Geometric property (T). Chin. Ann. Math. Ser. B 35(5) (2014), 761–800.CrossRefGoogle Scholar