Skip to main content Accessibility help
×
Home

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

  • Tobe Deprez (a1)

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.

Copyright

References

Hide All
1Adams, S.. Boundary amenability for word hyperbolic groups and an application to smooth dynamics of simple groups. Topology 33 (1994), 765783.
2Anantharaman-Delaroche, C.. Amenability and exactness for dynamical systems and their C*-algebras. Trans. Am. Math. Soc. 354 (2002), 41534179.
3Ando, H., Haagerup, U., Houdayer, C. and Marrakchi, A.. Structure of bicentralizer algebras and inclusions of type III factors (2018).
4Arzhantseva, G. and Delzant, T.. Examples of random groups (2008).
5Brodzki, J., Cave, C. and Li, K.. Exactness of locally compact groups. Adv. Math. 312 (2017), 209233.
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), 85125.
7Brown, N. and Ozawa, N.. C*-Algebras and Finite-Dimensional Approximations, volume 88 of Graduate Studies in Mathematics. American Mathematical Society, Providence, March 2008.
8Chifan, I. and Ioana, A.. Amalgamated free product rigidity for group von Neumann algebras. Adv. Math. 329 (2018), 819850.
9Chifan, I. and Sinclair, T.. On the structural theory of II1 factors of negatively curved groups. Ann. Sci. Ecole Norm. S. 46 (2013), 133.
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), 136159.
11Connes, A.. Classification of injective factors. Cases II1, II, IIIλ, λ ≠ 1. Ann. Math. 104 (1976), 73115.
12Cornulier, Y.. Locally compact wreath products. J. Aust. Math. Soc. (2017). https://www. cambridge.org/core/product/identifier/S1446788718000216/type/journal_article.
13Deprez, S. and Li, K.. Permanence properties of property A and coarse embeddability for locally compact groups (2014).
14Deprez, S. and Li, K.. Property A and uniform embedding for locally compact groups. J. Noncommut. Geom. 9 (2015), 797819.
15Feldman, J. and Moore, C. C.. Ergodic equivalence relations, cohomology, and von Neumann algebras. I. Trans. Am. Math. Soc. 234 (1977), 289324.
16Forrest, P.. On the virtual groups defined by ergodic actions of ℝn and ℤn. Adv. Math. 14 (1974), 271308.
17Gromov, M.. Random walk in random groups. Geom. Funct. Anal. 13 (2003), 73146.
18Guentner, E., Higson, N. and Weinberger, S.. The Novikov conjecture for linear groups. Publ. Math. Inst. Hautes Études Sci. 101 (2005), 243268.
19Haagerup, U. and Kraus, J.. Approximation properties for group C*-algebras and group von Neumann algebras. Trans. Am. Math. Soc. 344 (1994), 667699.
20Houdayer, C. and Isono, Y.. Unique prime factorization and bicentralizer problem for a class of type III factors. Adv. Math. 305 (2017), 402455.
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), 185219.
22Houdayer, C. and Vaes, S.. Type III factors with unique Cartan decomposition. J. Math. Pures Appl. 100 (2013), 564590.
23Kechris, A. S.. Classical Descriptive Set Theory, volume 156 of Graduate Texts in Mathematics (New York: Springer, 1995).
24Kirchberg, E. and Wassermann, S.. Exact groups and continuous bundles of C*-algebras. Math. Ann. 315 (1999), 169203.
25Kirchberg, E. and Wassermann, S.. Permanence properties of C*-exact groups. Doc. Math. 4 (1999), 513558.
26Koivisto, J., Kyed, D. and Raum, S.. Measure equivalence and coarse equivalence for unimodular locally compact groups (2017).
27Koivisto, J., Kyed, D. and Raum, S.. Measure equivalence for non-unimodular groups (2018).
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), 49174956.
29Mackey, G. W.. Induced representations of locally compact groups I. Ann. Math. 55 (1952), 101139.
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), 274294.
31Osajda, D.. Small cancellation labellings of some infinite graphs and applications (2014).
32Ozawa, N.. Solid von Neumann algebras. Acta Math. 192 (2004), 111117.
33Ozawa, N.. A Kurosh-type theorem for type II1 factors. Int. Math. Res. Not. 2006, 121.
34Ozawa, N.. An example of a solid von Neumann algebra. Hokkaido Math. J. 38 (2009), 557561.
35Ozawa, N. and Popa, S.. Some prime factorization results for type II1 factors. Invent. Math. 156 (2004), 223234.
36Popa, S. and Vaes, S.. Unique Cartan decomposition for II1 factors arising from arbitrary actions of hyperbolic groups. J. Reine Angew. Math. 694 (2014), 141198.
37Raum, S.. C*-simplicity after Breuillard, Haagerup, Kalantar, Kennedy and Ozawa (2019). http://raum-brothers.eu/sven/data/publications/bourbaki.pdf.
38Raum, S.. C*-simplicity of locally compact Powers groups. J. Reine Angew. Math. 748 (2019), 173205.
39Sako, H.. The class 𝒮 as an ME invariant. Int. Math. Res. Not., 2009, 27492759.
40Skandalis, G.. Une notion de nucléarité en K-théorie (d'après J. Cuntz). K-Theory 1 (1988), 549573.
41Struble, R. A.. Metrics in locally compact groups. Compos. Math. 28 (1974), 217222.
42Suzuki, Y.. Elementary constructions of non-discrete C*-simple groups. Proc. Am. Math. Soc. 145 (2016), 13691371.

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed