C*-simplicity and the unique trace property for discrete groups

Breuillard, Emmanuel; Kalantar, Mehrdad; Kennedy, Matthew; Ozawa, Narutaka

doi:10.1007/s10240-017-0091-2

Breuillard, Emmanuel ¹ ; Kalantar, Mehrdad ² ; Kennedy, Matthew ³ ; Ozawa, Narutaka ⁴

¹ Mathematisches Institut, Universität Münster 48149 Münster Germany
² Department of Mathematics, University of Houston 77204-3008 Houston TX United States
³ Department of Pure Mathematics, University of Waterloo N2L 3G1 Waterloo ON Canada
⁴ Research Institute for Mathematical Sciences, Kyoto University 606-8502 Kyoto Japan

Publications Mathématiques de l'IHÉS, Tome 126 (2017), pp. 35-71.

Résumé

A discrete group is said to be C*-simple if its reduced C*-algebra is simple, and is said to have the unique trace property if its reduced C*-algebra has a unique tracial state. A dynamical characterization of C*-simplicity was recently obtained by the second and third named authors. In this paper, we introduce new methods for working with group and crossed product C*-algebras that allow us to take the study of C*-simplicity a step further, and in addition to settle the longstanding open problem of characterizing groups with the unique trace property. We give a new and self-contained proof of the aforementioned characterization of C*-simplicity. This yields a new characterization of C*-simplicity in terms of the weak containment of quasi-regular representations. We introduce a convenient algebraic condition that implies C*-simplicity, and show that this condition is satisfied by a vast class of groups, encompassing virtually all previously known examples as well as many new ones. We also settle a question of Skandalis and de la Harpe on the simplicity of reduced crossed products. Finally, we introduce a new property for discrete groups that is closely related to C*-simplicity, and use it to prove a broad generalization of a theorem of Zimmer, originally conjectured by Connes and Sullivan, about amenable actions.

MR Zbl

DOI : 10.1007/s10240-017-0091-2

Affiliations des auteurs :

Breuillard, Emmanuel ¹ ; Kalantar, Mehrdad ² ; Kennedy, Matthew ³ ; Ozawa, Narutaka ⁴

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     author = {Breuillard, Emmanuel and Kalantar, Mehrdad and Kennedy, Matthew and Ozawa, Narutaka},
     title = {C*-simplicity and the unique trace property for discrete groups},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {35--71},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {126},
     year = {2017},
     doi = {10.1007/s10240-017-0091-2},
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     zbl = {1391.46071},
     language = {en},
     url = {http://www.numdam.org/articles/10.1007/s10240-017-0091-2/}
}

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Breuillard, Emmanuel; Kalantar, Mehrdad; Kennedy, Matthew; Ozawa, Narutaka. C*-simplicity and the unique trace property for discrete groups. Publications Mathématiques de l'IHÉS, Tome 126 (2017), pp. 35-71. doi : 10.1007/s10240-017-0091-2. http://www.numdam.org/articles/10.1007/s10240-017-0091-2/

Bibliographie
Cité par

[1.] Abért, M.; Glasner, Y.; Virág, B. Kesten’s theorem for invariant random subgroups, Duke Math. J., Volume 163 (2014), pp. 465-488 | DOI | MR | Zbl

[2.] Adyan, S. I. Random walks on free periodic groups, Izv. Math., Volume 21 (1983), pp. 425-434 | DOI | Zbl

[3.] Archbold, R. J.; Spielberg, J. S. Topologically free actions and ideals in discrete C*-dynamical systems, Proc. Edinb. Math. Soc., Volume 37 (1994), pp. 119-124 | DOI | MR | Zbl

[4.] Anantharaman-Delaroche, C. On spectral characterizations of amenability, Isr. J. Math., Volume 137 (2003), pp. 1-33 | DOI | MR | Zbl

[5.] Bader, U.; Duchesne, B.; Lecureux, J. Amenable invariant random subgroups, Isr. J. Math., Volume 213 (2016), pp. 399-422 | DOI | MR | Zbl

[6.] Bader, U.; Furman, A.; Sauer, R. Weak notions of normality and vanishing up to rank in $L^{2}$ -cohomology, Int. Math. Res. Not., Volume 12 (2014), pp. 3177-3189 | DOI | MR | Zbl

[7.] Bekka, M.; Cowling, M.; de la Harpe, P. Some groups whose reduced C*-algebra is simple, Publ. Math. Inst. Hautes Études Sci., Volume 80 (1994), pp. 117-134 | DOI | Numdam | MR | Zbl

[8.] E. Breuillard, A strong Tits alternative, | arXiv

[9.] Breuillard, E.; Gelander, T. A topological Tits alternative, Ann. Math., Volume 166 (2007), pp. 427-474 | DOI | MR | Zbl

[10.] Breuillard, E.; Gelander, T. Uniform independence in linear groups, Invent. Math., Volume 173 (2008), pp. 225-263 | DOI | MR | Zbl

[11.] Brown, N.; Ozawa, N. C*-Algebras and Finite-Dimensional Approximations (2008) | Zbl

[12.] Carrière, Y.; Ghys, É. Uniform independence in linear groups, C. R. Acad. Sci., Volume 300 (1985), pp. 677-680 | Zbl

[13.] Choi, M. D.; Effros, E. G. Injectivity and operator spaces, J. Funct. Anal., Volume 24 (1977), pp. 156-209 | DOI | MR | Zbl

[14.] Dahmani, F.; Guirardel, V.; Osin, D. Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, Mem. Am. Math. Soc., Volume 245 (2017) | MR | Zbl

[15.] Day, M. Amenable semigroups, Ill. J. Math., Volume 1 (1957), pp. 459-606 | MR | Zbl

[16.] de la Harpe, P. On simplicity of reduced C*-algebras of groups, Bull. Lond. Math. Soc., Volume 39 (2007), pp. 1-26 | DOI | MR | Zbl

[17.] de la Harpe, P.; Préaux, J. P. C*-simple groups: amalgamated free products, HNN extensions, and fundamental groups of 3-manifolds, J. Topol. Anal., Volume 3 (2011), pp. 451-489 | DOI | MR | Zbl

[18.] de La Harpe, P.; Skandalis, G. Powers’ property and simple C*-algebras, Math. Ann., Volume 273 (1986), pp. 241-250 | DOI | MR | Zbl

[19.] Dixmier, J. C*-Algebras (1977) | Zbl

[20.] Frolík, Z. Maps of extremally disconnected spaces, theory of types, and applications, General Topology and Its Relations to Modern Analysis and Algebra (1971), pp. 131-142

[21.] Furman, A. On minimal strongly proximal actions of locally compact groups, Isr. J. Math., Volume 136 (2003), pp. 173-187 | DOI | MR | Zbl

[22.] Furstenberg, H. Boundary Theory and Stochastic Processes on Homogeneous Spaces (1973) | Zbl

[23.] Gaboriau, D. Coût des relations d’équivalence et des groupes, Invent. Math., Volume 139 (2000), pp. 41-98 | DOI | MR | Zbl

[24.] Glasner, S. Topological dynamics and group theory, Trans. Am. Math. Soc., Volume 187 (1974), pp. 327-334 | DOI | MR | Zbl

[25.] Glasner, S. Proximal Flows (1976) | Zbl

[26.] Gleason, A. M. Projective topological spaces, Ill. J. Math., Volume 2 (1958), pp. 482-489 | MR | Zbl

[27.] U. Haagerup, A new look at C*-simplicity and the unique trace property of a group, | arXiv

[28.] U. Haagerup and K. K. Olesen, Non-inner amenability of the Thompson groups T and V, | arXiv

[29.] Hamana, M. Injective envelopes of C*-dynamical systems, Tohoku Math. J., Volume 37 (1985), pp. 463-487 | DOI | MR | Zbl

[30.] Howe, R.; Tan, E-C. Nonabelian Harmonic Analysis (1992)

[31.] Hull, M.; Osin, D. Induced quasicocycles on groups with hyperbolically embedded subgroups, Algebraic Geom. Topol., Volume 13 (2013), pp. 2635-2665 | DOI | MR | Zbl

[32.] Ivanov, S. V. The free Burnside groups of sufficiently large exponents, Int. J. Algebra Comput., Volume 4 (1994), pp. 1-308 | DOI | MR | Zbl

[33.] Kawamura, S.; Tomiyama, J. Properties of topological dynamical systems and corresponding C*-algebras, Tokyo J. Math., Volume 13 (1990), pp. 215-257 | DOI | MR | Zbl

[34.] M. Kennedy, An intrinsic characterization of C*-simplicity, | arXiv

[35.] Kalantar, M.; Kennedy, M. Boundaries of reduced C*-algebras of discrete groups, J. Reine Angew. Math., Volume 727 (2017), pp. 247-267 | MR | Zbl

[36.] Kuhn, M. G. Amenable actions and weak containment of certain representations of discrete groups, Proc. Am. Math. Soc., Volume 122 (1994), pp. 751-757 | DOI | MR | Zbl

[37.] A. Le Boudec, C*-simplicity and the amenable radical, | arXiv

[38.] A. Le Boudec and N. Matte Bon, Subgroup dynamics and C*-simplicity of groups of homeomorphisms, | arXiv

[39.] Lück, W. Dimension theory of arbitrary modules over finite von Neumann algebras and $L^{2}$ -Betti numbers, I: foundations, J. Reine Angew. Math., Volume 495 (1998), pp. 135-162 | MR | Zbl

[40.] Monod, N. Continuous Bounded Cohomology of Locally Compact Groups (2001) | DOI | Zbl

[41.] N. Monod and Y. Shalom, Orbit equivalence rigidity and bounded cohomology, Ann. Math. (2006), 825–878.

[42.] Olshanskii, A. Y. On the question of the existence of an invariant mean on a group, Usp. Mat. Nauk, Volume 35 (1980), pp. 199-200 | MR

[43.] Olshanskii, A. Y. Geometry of Defining Relations in Groups (1991) | DOI

[44.] Olshanskii, A. Y.; Osin, D. V. C*-simple groups without free subgroups, Groups Geom. Dyn., Volume 8 (2014), pp. 93-983 | DOI | MR | Zbl

[45.] Osin, D. V. Acylindrically hyperbolic groups, Trans. Am. Math. Soc., Volume 368 (2016), pp. 851-888 | DOI | MR | Zbl

[46.] Paulsen, V. Completely Bounded Maps and Operator Algebras (2002) | Zbl

[47.] D. Pitts, Structure for regular inclusions, | arXiv

[48.] Powers, R. Simplicity of the C*-algebra associated with the free group on two generators, Duke Math. J., Volume 42 (1975), pp. 151-156 | DOI | MR | Zbl

[49.] T. Poznansky, Characterization of linear groups whose reduced C*-algebras are simple, | arXiv

[50.] Thom, A. Low degree bounded cohomology and $L^{2}$ -invariants for negatively curved groups, Groups Geom. Dyn., Volume 3 (2009), pp. 343-358 | DOI | MR | Zbl

[51.] Tits, J. Free subgroups in linear groups, J. Algebra, Volume 20 (1972), pp. 250-270 | DOI | MR | Zbl

[52.] R. D. Tucker-Drob, Shift-minimal groups, fixed price 1, and the unique trace property, | arXiv

[53.] Zimmer, R. J. Amenable actions and dense subgroups of Lie groups, J. Funct. Anal., Volume 72 (1987), pp. 58-64 | DOI | MR | Zbl

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