Amenability versus property $(T)$ for non-locally compact topological groups
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- by Vladimir G. Pestov PDF
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Abstract:
For locally compact groups amenability and Kazhdan’s property $(T)$ are mutually exclusive in the sense that a group having both properties is compact. This is no longer true for more general Polish groups. However, a weaker result still holds for SIN groups (topological groups admitting a basis of conjugation-invariant neighbourhoods of identity): if such a group admits sufficiently many unitary representations, then it is precompact as soon as it is amenable and has the strong property $(T)$ (i.e., admits a finite Kazhdan set). If an amenable topological group with property $(T)$ admits a faithful uniformly continuous representation, then it is maximally almost periodic. In particular, an extremely amenable SIN group never has strong property $(T)$, and an extremely amenable subgroup of unitary operators in the uniform topology is never a Kazhdan group. This leads to first examples distinguishing between property $(T)$ and property $(FH)$ in the class of Polish groups. Disproving a 2003 conjecture by Bekka, we construct a complete, separable, minimally almost periodic topological group with property $(T)$ having no finite Kazhdan set. Finally, as a curiosity, we observe that the class of topological groups with property $(T)$ is closed under arbitrary infinite products with the usual product topology.References
- C. J. Atkin, The Finsler geometry of certain covering groups of operator groups, Hokkaido Math. J. 18 (1989), no. 1, 45–77. MR 985431, DOI 10.14492/hokmj/1381517780
- C. J. Atkin, Boundedness in uniform spaces, topological groups, and homogeneous spaces, Acta Math. Hungar. 57 (1991), no. 3-4, 213–232. MR 1139315, DOI 10.1007/BF01903672
- Howard Becker and Alexander S. Kechris, The descriptive set theory of Polish group actions, London Mathematical Society Lecture Note Series, vol. 232, Cambridge University Press, Cambridge, 1996. MR 1425877, DOI 10.1017/CBO9780511735264
- Mohammed E. B. Bekka, Amenable unitary representations of locally compact groups, Invent. Math. 100 (1990), no. 2, 383–401. MR 1047140, DOI 10.1007/BF01231192
- M. B. Bekka, Kazhdan’s property (T) for the unitary group of a separable Hilbert space, Geom. Funct. Anal. 13 (2003), no. 3, 509–520. MR 1995797, DOI 10.1007/s00039-003-0420-0
- Bachir Bekka, Pierre de la Harpe, and Alain Valette, Kazhdan’s property (T), New Mathematical Monographs, vol. 11, Cambridge University Press, Cambridge, 2008. MR 2415834, DOI 10.1017/CBO9780511542749
- Mohammed E. B. Bekka and Alain Valette, Kazhdan’s property $(\textrm {T})$ and amenable representations, Math. Z. 212 (1993), no. 2, 293–299. MR 1202813, DOI 10.1007/BF02571659
- George M. Bergman, Generating infinite symmetric groups, Bull. London Math. Soc. 38 (2006), no. 3, 429–440. MR 2239037, DOI 10.1112/S0024609305018308
- Valerio Capraro and Martino Lupini, Introduction to sofic and hyperlinear groups and Connes’ embedding conjecture, Lecture Notes in Mathematics, vol. 2136, Springer, Cham, 2015. With an appendix by Vladimir Pestov. MR 3408561, DOI 10.1007/978-3-319-19333-5
- Pierre-Alain Cherix, Michael Cowling, and Bernd Straub, Filter products of $C_0$-semigroups and ultraproduct representations for Lie groups, J. Funct. Anal. 208 (2004), no. 1, 31–63. MR 2034291, DOI 10.1016/j.jfa.2003.06.007
- A. Connes, Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not =1$, Ann. of Math. (2) 104 (1976), no. 1, 73–115. MR 454659, DOI 10.2307/1971057
- A. Connes and V. Jones, Property $T$ for von Neumann algebras, Bull. London Math. Soc. 17 (1985), no. 1, 57–62. MR 766450, DOI 10.1112/blms/17.1.57
- Yves de Cornulier, Strongly bounded groups and infinite powers of finite groups, Comm. Algebra 34 (2006), no. 7, 2337–2345. MR 2240370, DOI 10.1080/00927870600550194
- Pierre de la Harpe, Moyennabilité de quelques groupes topologiques de dimension infinie, C. R. Acad. Sci. Paris Sér. A-B 277 (1973), A1037–A1040 (French). MR 333060
- P. de la Harpe, Moyennabilité du groupe unitaire et propriété $P$ de Schwartz des algèbres de von Neumann, Algèbres d’opérateurs (Sém., Les Plans-sur-Bex, 1978) Lecture Notes in Math., vol. 725, Springer, Berlin, 1979, pp. 220–227 (French). MR 548116
- Patrick Delorme, $1$-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles. Produits tensoriels continus de représentations, Bull. Soc. Math. France 105 (1977), no. 3, 281–336 (French). MR 578893, DOI 10.24033/bsmf.1853
- Manfred Droste and W. Charles Holland, Generating automorphism groups of chains, Forum Math. 17 (2005), no. 4, 699–710. MR 2154425, DOI 10.1515/form.2005.17.4.699
- Manfred Droste, W. Charles Holland, and Georg Ulbrich, On full groups of measure-preserving and ergodic transformations with uncountable cofinalities, Bull. Lond. Math. Soc. 40 (2008), no. 3, 463–472. MR 2418802, DOI 10.1112/blms/bdn028
- H. A. Dye, On groups of measure preserving transformations. I, Amer. J. Math. 81 (1959), 119–159. MR 131516, DOI 10.2307/2372852
- Ilijas Farah and Sławomir Solecki, Extreme amenability of $L_0$, a Ramsey theorem, and Lévy groups, J. Funct. Anal. 255 (2008), no. 2, 471–493. MR 2419967, DOI 10.1016/j.jfa.2008.03.016
- Stanley P. Franklin and Barbara V. Smith Thomas, A survey of $k_{\omega }$-spaces, Topology Proc. 2 (1977), no. 1, 111–124 (1978). MR 540599
- Thierry Giordano and Vladimir Pestov, Some extremely amenable groups related to operator algebras and ergodic theory, J. Inst. Math. Jussieu 6 (2007), no. 2, 279–315. MR 2311665, DOI 10.1017/S1474748006000090
- Eli Glasner, On minimal actions of Polish groups, Topology Appl. 85 (1998), no. 1-3, 119–125. 8th Prague Topological Symposium on General Topology and Its Relations to Modern Analysis and Algebra (1996). MR 1617456, DOI 10.1016/S0166-8641(97)00143-0
- E. Glasner, B. Tsirelson, and B. Weiss, The automorphism group of the Gaussian measure cannot act pointwise, Israel J. Math. 148 (2005), 305–329. Probability in mathematics. MR 2191233, DOI 10.1007/BF02775441
- E. Glasner and B. Weiss, Minimal actions of the group $\Bbb S(\Bbb Z)$ of permutations of the integers, Geom. Funct. Anal. 12 (2002), no. 5, 964–988. MR 1937832, DOI 10.1007/PL00012651
- M. I. Graev, Theory of topological groups. I. Norms and metrics on groups. Complete groups. Free topological groups, Uspehi Matem. Nauk (N.S.) 5 (1950), no. 2(36), 3–56 (Russian). MR 0036245
- M. Gromov and V. D. Milman, A topological application of the isoperimetric inequality, Amer. J. Math. 105 (1983), no. 4, 843–854. MR 708367, DOI 10.2307/2374298
- Alain Guichardet, Sur la cohomologie des groupes topologiques. II, Bull. Sci. Math. (2) 96 (1972), 305–332 (French). MR 340464
- U. Haagerup, All nuclear $C^{\ast }$-algebras are amenable, Invent. Math. 74 (1983), no. 2, 305–319. MR 723220, DOI 10.1007/BF01394319
- Jan Hejcman, Boundedness in uniform spaces and topological groups, Czechoslovak Math. J. 9(84) (1959), 544–563 (English, with Russian summary). MR 141075, DOI 10.21136/CMJ.1959.100381
- Wojchiech Herer and Jens Peter Reus Christensen, On the existence of pathological submeasures and the construction of exotic topological groups, Math. Ann. 213 (1975), 203–210. MR 412369, DOI 10.1007/BF01350870
- V. F. R. Jones, Von Neumann algebras, lecture notes, version of 13th May 2003, 121 pp., http://www.imsc.res.in/$\sim$vpgupta/hnotes/notes-jones.pdf
- Alexander S. Kechris and Christian Rosendal, Turbulence, amalgamation, and generic automorphisms of homogeneous structures, Proc. Lond. Math. Soc. (3) 94 (2007), no. 2, 302–350. MR 2308230, DOI 10.1112/plms/pdl007
- Arthur Lieberman, The structure of certain unitary representations of infinite symmetric groups, Trans. Amer. Math. Soc. 164 (1972), 189–198. MR 286940, DOI 10.1090/S0002-9947-1972-0286940-2
- Michael G. Megrelishvili, Every semitopological semigroup compactification of the group $H_+[0,1]$ is trivial, Semigroup Forum 63 (2001), no. 3, 357–370. MR 1851816, DOI 10.1007/s002330010076
- Narutaka Ozawa, There is no separable universal $\rm II_1$-factor, Proc. Amer. Math. Soc. 132 (2004), no. 2, 487–490. MR 2022373, DOI 10.1090/S0002-9939-03-07127-2
- Alan L. T. Paterson, Amenability, Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, RI, 1988. MR 961261, DOI 10.1090/surv/029
- Vladimir G. Pestov, On free actions, minimal flows, and a problem by Ellis, Trans. Amer. Math. Soc. 350 (1998), no. 10, 4149–4165. MR 1608494, DOI 10.1090/S0002-9947-98-02329-0
- V. G. Pestov, Amenable representations and dynamics of the unit sphere in an infinite-dimensional Hilbert space, Geom. Funct. Anal. 10 (2000), no. 5, 1171–1201. MR 1800066, DOI 10.1007/PL00001650
- Vladimir Pestov, Ramsey-Milman phenomenon, Urysohn metric spaces, and extremely amenable groups, Israel J. Math. 127 (2002), 317–357. MR 1900705, DOI 10.1007/BF02784537
- Vladimir Pestov, Dynamics of infinite-dimensional groups, University Lecture Series, vol. 40, American Mathematical Society, Providence, RI, 2006. The Ramsey-Dvoretzky-Milman phenomenon; Revised edition of Dynamics of infinite-dimensional groups and Ramsey-type phenomena [Inst. Mat. Pura. Apl. (IMPA), Rio de Janeiro, 2005; MR2164572]. MR 2277969, DOI 10.1090/ulect/040
- Sorin Popa and Masamichi Takesaki, The topological structure of the unitary and automorphism groups of a factor, Comm. Math. Phys. 155 (1993), no. 1, 93–101. MR 1228527, DOI 10.1007/BF02100051
- Walter Roelcke and Susanne Dierolf, Uniform structures on topological groups and their quotients, Advanced Book Program, McGraw-Hill International Book Co., New York, 1981. MR 644485
- Christian Rosendal, A topological version of the Bergman property, Forum Math. 21 (2009), no. 2, 299–332. MR 2503307, DOI 10.1515/FORUM.2009.014
- Christian Rosendal, Global and local boundedness of Polish groups, Indiana Univ. Math. J. 62 (2013), no. 5, 1621–1678. MR 3188557, DOI 10.1512/iumj.2013.62.5133
- Éric Ricard and Christian Rosendal, On the algebraic structure of the unitary group, Collect. Math. 58 (2007), no. 2, 181–192. MR 2332091
- Marcin Sabok, Extreme amenability of abelian $L_0$ groups, J. Funct. Anal. 263 (2012), no. 10, 2978–2992. MR 2973332, DOI 10.1016/j.jfa.2012.07.011
- Sławomir Solecki, Unitary representations of the groups of measurable and continuous functions with values in the circle, J. Funct. Anal. 267 (2014), no. 9, 3105–3124. MR 3261106, DOI 10.1016/j.jfa.2014.08.003
- Todor Tsankov, Unitary representations of oligomorphic groups, Geom. Funct. Anal. 22 (2012), no. 2, 528–555. MR 2929072, DOI 10.1007/s00039-012-0156-9
- Alain Valette, Amenable representations and finite injective von Neumann algebras, Proc. Amer. Math. Soc. 125 (1997), no. 6, 1841–1843. MR 1371145, DOI 10.1090/S0002-9939-97-03754-4
Additional Information
- Vladimir G. Pestov
- Affiliation: Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, Ontario, Canada K1N 6N5 – and – Departamento de Matemática, Universidade Federal de Santa Catarina, Trindade, Florianópolis, SC, 88.040-900, Brazil
- MR Author ID: 138420
- Email: vpest283@uottawa.ca
- Received by editor(s): January 31, 2017
- Received by editor(s) in revised form: April 4, 2017
- Published electronically: July 5, 2018
- Additional Notes: The author was Special Visiting Researcher of the program Science Without Borders of CAPES (Brazil), processo 085/2012.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 7417-7436
- MSC (2010): Primary 22A25, 43A65, 57S99
- DOI: https://doi.org/10.1090/tran/7256
- MathSciNet review: 3841853