Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-12-01T03:00:56.811Z Has data issue: false hasContentIssue false
  • English
  • Français

Ozawa's class 𝒮 for locally compact groups and unique prime factorization of group von Neumann algebras

Published online by Cambridge University Press:  19 July 2019

Tobe Deprez
Tobe Deprez*
Affiliation:
Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3001Leuven, Belgium ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We study class 𝒮 for locally compact groups. We characterize locally compact groups in this class as groups having an amenable action on a boundary that is small at infinity, generalizing a theorem of Ozawa. Using this characterization, we provide new examples of groups in class 𝒮 and prove a unique prime factorization theorem for group von Neumann algebras of products of locally compact groups in this class. We also prove that class 𝒮 is a measure equivalence invariant.

Keywords

locally compact groupsvon Neumann algebraclass 𝒮unique prime factorizationmeasure equivalence
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2656 - 2681
Check for updates
Copyright
Copyright © Royal Society of Edinburgh 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Adams, S.. Boundary amenability for word hyperbolic groups and an application to smooth dynamics of simple groups. Topology 33 (1994), 765–783.CrossRefGoogle Scholar
2Anantharaman-Delaroche, C.. Amenability and exactness for dynamical systems and their C*-algebras. Trans. Am. Math. Soc. 354 (2002), 4153–4179.CrossRefGoogle Scholar
3Ando, H., Haagerup, U., Houdayer, C. and Marrakchi, A.. Structure of bicentralizer algebras and inclusions of type III factors (2018).Google Scholar
4Arzhantseva, G. and Delzant, T.. Examples of random groups (2008).Google Scholar
5Brodzki, J., Cave, C. and Li, K.. Exactness of locally compact groups. Adv. Math. 312 (2017), 209–233.CrossRefGoogle Scholar
6Brothier, A., Deprez, T. and Vaes, S.. Rigidity for von Neumann algebras given by locally compact groups and their crossed products. Commun. Math. Phys. 361 (2018), 85–125.CrossRefGoogle Scholar
7Brown, N. and Ozawa, N.. C*-Algebras and Finite-Dimensional Approximations, volume 88 of Graduate Studies in Mathematics. American Mathematical Society, Providence, March 2008.Google Scholar
8Chifan, I. and Ioana, A.. Amalgamated free product rigidity for group von Neumann algebras. Adv. Math. 329 (2018), 819–850.CrossRefGoogle Scholar
9Chifan, I. and Sinclair, T.. On the structural theory of II1 factors of negatively curved groups. Ann. Sci. Ecole Norm. S. 46 (2013), 1–33.CrossRefGoogle Scholar
10Chifan, I., de Santiago, R. and Sinclair, T.. W*-rigidity for the von Neumann algebras of products of hyperbolic groups. Geom. Funct. Anal. 26 (2016), 136–159.CrossRefGoogle Scholar
11Connes, A.. Classification of injective factors. Cases II1, II∞, IIIλ, λ ≠ 1. Ann. Math. 104 (1976), 73–115.CrossRefGoogle Scholar
12Cornulier, Y.. Locally compact wreath products. J. Aust. Math. Soc. (2017). https://www. cambridge.org/core/product/identifier/S1446788718000216/type/journal_article.Google Scholar
13Deprez, S. and Li, K.. Permanence properties of property A and coarse embeddability for locally compact groups (2014).Google Scholar
14Deprez, S. and Li, K.. Property A and uniform embedding for locally compact groups. J. Noncommut. Geom. 9 (2015), 797–819.CrossRefGoogle Scholar
15Feldman, J. and Moore, C. C.. Ergodic equivalence relations, cohomology, and von Neumann algebras. I. Trans. Am. Math. Soc. 234 (1977), 289–324.CrossRefGoogle Scholar
16Forrest, P.. On the virtual groups defined by ergodic actions of ℝn and ℤn. Adv. Math. 14 (1974), 271–308.CrossRefGoogle Scholar
17Gromov, M.. Random walk in random groups. Geom. Funct. Anal. 13 (2003), 73–146.CrossRefGoogle Scholar
18Guentner, E., Higson, N. and Weinberger, S.. The Novikov conjecture for linear groups. Publ. Math. Inst. Hautes Études Sci. 101 (2005), 243–268.CrossRefGoogle Scholar
19Haagerup, U. and Kraus, J.. Approximation properties for group C*-algebras and group von Neumann algebras. Trans. Am. Math. Soc. 344 (1994), 667–699.Google Scholar
20Houdayer, C. and Isono, Y.. Unique prime factorization and bicentralizer problem for a class of type III factors. Adv. Math. 305 (2017), 402–455.CrossRefGoogle Scholar
21Houdayer, C. and Raum, S.. Locally compact groups acting on trees, the type I conjecture and non-amenable von Neumann algebras. Comment. Math. Helv. 94 (2019), 185–219.CrossRefGoogle Scholar
22Houdayer, C. and Vaes, S.. Type III factors with unique Cartan decomposition. J. Math. Pures Appl. 100 (2013), 564–590.CrossRefGoogle Scholar
23Kechris, A. S.. Classical Descriptive Set Theory, volume 156 of Graduate Texts in Mathematics (New York: Springer, 1995).CrossRefGoogle Scholar
24Kirchberg, E. and Wassermann, S.. Exact groups and continuous bundles of C*-algebras. Math. Ann. 315 (1999), 169–203.CrossRefGoogle Scholar
25Kirchberg, E. and Wassermann, S.. Permanence properties of C*-exact groups. Doc. Math. 4 (1999), 513–558.Google Scholar
26Koivisto, J., Kyed, D. and Raum, S.. Measure equivalence and coarse equivalence for unimodular locally compact groups (2017).Google Scholar
27Koivisto, J., Kyed, D. and Raum, S.. Measure equivalence for non-unimodular groups (2018).Google Scholar
28Kyed, D., Petersen, H. D. and Vaes, S.. L 2-Betti numbers of locally compact groups and their cross section equivalence relations. Trans. Am. Math. Soc. 367 (2015), 4917–4956.CrossRefGoogle Scholar
29Mackey, G. W.. Induced representations of locally compact groups I. Ann. Math. 55 (1952), 101–139.CrossRefGoogle Scholar
30Meesschaert, N., Raum, S. and Vaes, S.. Stable orbit equivalence of Bernoulli actions of free groups and isomorphism of some of their factor actions. Expo. Math. 31 (2013), 274–294.CrossRefGoogle Scholar
31Osajda, D.. Small cancellation labellings of some infinite graphs and applications (2014).Google Scholar
32Ozawa, N.. Solid von Neumann algebras. Acta Math. 192 (2004), 111–117.CrossRefGoogle Scholar
33Ozawa, N.. A Kurosh-type theorem for type II1 factors. Int. Math. Res. Not. 2006, 1–21.Google Scholar
34Ozawa, N.. An example of a solid von Neumann algebra. Hokkaido Math. J. 38 (2009), 557–561.CrossRefGoogle Scholar
35Ozawa, N. and Popa, S.. Some prime factorization results for type II1 factors. Invent. Math. 156 (2004), 223–234.CrossRefGoogle Scholar
36Popa, S. and Vaes, S.. Unique Cartan decomposition for II1 factors arising from arbitrary actions of hyperbolic groups. J. Reine Angew. Math. 694 (2014), 141–198.Google Scholar
37Raum, S.. C*-simplicity after Breuillard, Haagerup, Kalantar, Kennedy and Ozawa (2019). http://raum-brothers.eu/sven/data/publications/bourbaki.pdf.Google Scholar
38Raum, S.. C*-simplicity of locally compact Powers groups. J. Reine Angew. Math. 748 (2019), 173–205.CrossRefGoogle Scholar
39Sako, H.. The class 𝒮 as an ME invariant. Int. Math. Res. Not., 2009, 2749–2759.Google Scholar
40Skandalis, G.. Une notion de nucléarité en K-théorie (d'après J. Cuntz). K-Theory 1 (1988), 549–573.CrossRefGoogle Scholar
41Struble, R. A.. Metrics in locally compact groups. Compos. Math. 28 (1974), 217–222.Google Scholar
42Suzuki, Y.. Elementary constructions of non-discrete C*-simple groups. Proc. Am. Math. Soc. 145 (2016), 1369–1371.CrossRefGoogle Scholar