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.

^{1}; Kalantar, Mehrdad

^{2}; Kennedy, Matthew

^{3}; Ozawa, Narutaka

^{4}

@article{PMIHES_2017__126__35_0, 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}, mrnumber = {3735864}, zbl = {1391.46071}, language = {en}, url = {http://www.numdam.org/articles/10.1007/s10240-017-0091-2/} }

TY - JOUR AU - Breuillard, Emmanuel AU - Kalantar, Mehrdad AU - Kennedy, Matthew AU - Ozawa, Narutaka TI - C*-simplicity and the unique trace property for discrete groups JO - Publications Mathématiques de l'IHÉS PY - 2017 SP - 35 EP - 71 VL - 126 PB - Springer Berlin Heidelberg PP - Berlin/Heidelberg UR - http://www.numdam.org/articles/10.1007/s10240-017-0091-2/ DO - 10.1007/s10240-017-0091-2 LA - en ID - PMIHES_2017__126__35_0 ER -

%0 Journal Article %A Breuillard, Emmanuel %A Kalantar, Mehrdad %A Kennedy, Matthew %A Ozawa, Narutaka %T C*-simplicity and the unique trace property for discrete groups %J Publications Mathématiques de l'IHÉS %D 2017 %P 35-71 %V 126 %I Springer Berlin Heidelberg %C Berlin/Heidelberg %U http://www.numdam.org/articles/10.1007/s10240-017-0091-2/ %R 10.1007/s10240-017-0091-2 %G en %F PMIHES_2017__126__35_0

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

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

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

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

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

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

[6.] 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.] 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.] A topological Tits alternative, Ann. Math., Volume 166 (2007), pp. 427-474 | DOI | MR | Zbl

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

[11.] C*-Algebras and Finite-Dimensional Approximations (2008) | Zbl

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

[13.] Injectivity and operator spaces, J. Funct. Anal., Volume 24 (1977), pp. 156-209 | DOI | MR | Zbl

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

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

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

[17.] 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.] Powers’ property and simple C*-algebras, Math. Ann., Volume 273 (1986), pp. 241-250 | DOI | MR | Zbl

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

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

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

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

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

[25.] Proximal Flows (1976) | Zbl

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

[30.] Nonabelian Harmonic Analysis (1992)

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

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

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

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

[43.] Geometry of Defining Relations in Groups (1991) | DOI

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

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

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

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

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

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