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PROPERTY (T) AND STRONG 1-BOUNDEDNESS FOR VON NEUMANN ALGEBRAS
Part of:
Selfadjoint operator algebras
Locally compact groups and their algebras
Normed linear spaces and Banach spaces; Banach lattices
Published online by Cambridge University Press: 01 September 2025
Abstract
The notion of strong 1-boundedness for finite von Neumann algebras was introduced in [Jun07b]. This framework provided a free probabilistic approach to study rigidity properties and classification of finite von Neumann algebras. In this paper, we prove that tracial von Neumann algebras with a finite Kazhdan set are strongly 1-bounded. This includes all property (T) von Neumann algebras with finite-dimensional center and group von Neumann algebras of property (T) groups. This result generalizes all the previous results in this direction due to Voiculescu, Ge, Ge-Shen, Connes-Shlyakhtenko, Jung-Shlyakhtenko, Jung and Shlyakhtenko. Our proofs are based on analysis of covering estimates of microstate spaces using an iteration technique in the spirit of Jung.
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Footnotes
B. Hayes gratefully acknowledges support from the NSF grant DMS-2000105. D. Jekel was supported by NSF grant DMS-2002826
References
Anderson, GW, Guionnet, A and Zeitouni, O (2009) An Introduction to Random Matrices, Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010. xiv+492 pp.10.1017/CBO9780511801334CrossRefGoogle Scholar
Atkinson, S and Kunnawalkam, Elayavalli S (2021) On ultraproduct embeddings and amenability for tracial von Neumann algebras. Int. Math. Res. Not. IMRN (4), 2882–2918. MR 421834110.1093/imrn/rnaa257CrossRefGoogle Scholar
Brown, NP, Dykema, KJ and Jung, K (2008) Free entropy dimension in amalgamated free products. Proceedings of the London Mathematical Society 97(2), 339–367.10.1112/plms/pdm054CrossRefGoogle Scholar
Bekka, B., De la Harpe, P and Valette, A (2008) Kazhdan’s Property (T), New Mathematical Monographs, vol. 11. Cambridge: Cambridge University Press. MR 241583410.1017/CBO9780511542749CrossRefGoogle Scholar
Brown, NP and Ozawa, N (2008)
${C}^{\ast }$
-Algebras and Finite-Dimensional Approximations, Graduate Studies in Mathematics, vol. 88. Providence: American Mathematical Society.10.1090/gsm/088CrossRefGoogle Scholar
${C}^{\ast }$
-Algebras and Finite-Dimensional Approximations, Graduate Studies in Mathematics, vol. 88. Providence: American Mathematical Society.10.1090/gsm/088CrossRefGoogle ScholarBrannan, M and Vergnioux, R (2018) Orthogonal free quantum group factors are strongly 1-bounded. Adv. Math. 329, 133–156. MR 378341010.1016/j.aim.2018.02.007CrossRefGoogle Scholar
Chifan, I, Das, S, Houdayer, C and Khan, K Examples of property (T)
$\mathrm{I}{\mathrm{I}}_1$
factors with trivial fundamental group. American Jounral of Math. 146 (2024), no. 2, 435–465.Google Scholar
$\mathrm{I}{\mathrm{I}}_1$
factors with trivial fundamental group. American Jounral of Math. 146 (2024), no. 2, 435–465.Google ScholarConnes, A. and Jones, V (1985) Property
$\mathrm{T}$
for von Neumann algebras. Bull. London Math. Soc. 17(1), 57–62. MR 76645010.1112/blms/17.1.57CrossRefGoogle Scholar
$\mathrm{T}$
for von Neumann algebras. Bull. London Math. Soc. 17(1), 57–62. MR 76645010.1112/blms/17.1.57CrossRefGoogle ScholarConnes, A (1976) Classification of injective factors. Cases
$\mathrm{I}{\mathrm{I}}_1,$
$\mathrm{I}{\mathrm{I}}_{\infty },$
$\mathrm{II}{\mathrm{I}}_{\lambda},$
$\lambda \ne 1$
. Ann. of Math. (2) 104(1), 73–115. MR 045465910.2307/1971057CrossRefGoogle Scholar
$\mathrm{I}{\mathrm{I}}_1,$
$\mathrm{I}{\mathrm{I}}_{\infty },$
$\mathrm{II}{\mathrm{I}}_{\lambda},$
$\lambda \ne 1$
. Ann. of Math. (2) 104(1), 73–115. MR 045465910.2307/1971057CrossRefGoogle ScholarConnes, A (1980) A factor of type
$\mathrm{I}{\mathrm{I}}_1$
with countable fundamental group. J. Operator Theory 4(1), 151–153. MR 587372Google Scholar
$\mathrm{I}{\mathrm{I}}_1$
with countable fundamental group. J. Operator Theory 4(1), 151–153. MR 587372Google ScholarConnes, A (1982) Classification des facteurs. In Operator Algebras and Applications, Part 2 (Kingston, Ont., 1980), Proc. Sympos. Pure Math., vol. 38. Providence, RI: Amer. Math. Soc., 43–109. MR 67949710.1090/pspum/038.2/679497CrossRefGoogle Scholar
Conway, JB (1990) A Course in Functional Analysis, 2nd edn., Graduate Texts in Mathematics, vol. 96. New York: Springer-Verlag. MR 1070713Google Scholar
Connes, A and Shlyakhtenko, D (2005)
${L}^2$
-homology for von Neumann algebras. J. Reine Angew. Math. 586, 125–168. MR 218060310.1515/crll.2005.2005.586.125CrossRefGoogle Scholar
${L}^2$
-homology for von Neumann algebras. J. Reine Angew. Math. 586, 125–168. MR 218060310.1515/crll.2005.2005.586.125CrossRefGoogle ScholarConnes, A and Weiss, B (1980) Property and asymptotically invariant sequences. Israel J. Math. 37(3), 209–210. MR 59945510.1007/BF02760962CrossRefGoogle Scholar
Delorme, P (1977)
$1$
-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles. Produits tensoriels continus de representations. Bull. Soc. Math. France 105(3), 281–336. MR 57889310.24033/bsmf.1853CrossRefGoogle Scholar
$1$
-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles. Produits tensoriels continus de representations. Bull. Soc. Math. France 105(3), 281–336. MR 57889310.24033/bsmf.1853CrossRefGoogle ScholarDixmier, J (1953) Formes linéaires sur un anneau d’opérateurs. Bulletin de la Société Mathématique de France 81, 9–39.10.24033/bsmf.1436CrossRefGoogle Scholar
Dykema, K, Jung, K and Shlyakhtenko, D (2005) The microstates free entropy dimension of any DT-operator is 2. Doc. Math. 10, 247–261. MR 214807610.4171/dm/188CrossRefGoogle Scholar
De Santiago, R, Hayes, B, Hoff, D and Sinclair, T (to appear) Maximal rigid subalgebras of deformations and
${\mathrm{l}}^2$
-cohomology, Anal. PDE.Google Scholar
${\mathrm{l}}^2$
-cohomology, Anal. PDE.Google ScholarDykema, K (1994) Interpolated free group factors. Pacific J. Math. 163(1), 123–135. MR 125617910.2140/pjm.1994.163.123CrossRefGoogle Scholar
Elek, G and Szabó, E (2005) Hyperlinearity, essentially free actions and
${L}^2$
-invariants. The sofic property. Math. Ann. 332(2), 421–441. MR 217806910.1007/s00208-005-0640-8CrossRefGoogle Scholar
${L}^2$
-invariants. The sofic property. Math. Ann. 332(2), 421–441. MR 217806910.1007/s00208-005-0640-8CrossRefGoogle ScholarFurman, A (1999) Gromov’s measure equivalence and rigidity of higher rank lattices. Ann. of Math. (2) 150(3), 1059–1081. MR 174098610.2307/121062CrossRefGoogle Scholar
Furman, A (1999) Orbit equivalence rigidity. Ann. of Math. (2) 150(3), 1083–1108. MR 174098510.2307/121063CrossRefGoogle Scholar
Gaboriau, D (2000) Coût des relations d’équivalence et des groups. Invent. Math. 139(1), 41–98. MR 172887610.1007/s002229900019CrossRefGoogle Scholar
Gaboriau, D (2002) Invariants
${l}^2$
de relations d’équivalence et de groups. Publ. Math. Inst. Hautes Études Sci. (95), 93–150. MR 195319110.1007/s102400200002CrossRefGoogle Scholar
${l}^2$
de relations d’équivalence et de groups. Publ. Math. Inst. Hautes Études Sci. (95), 93–150. MR 195319110.1007/s102400200002CrossRefGoogle ScholarGaboriau, D (2005) Invariant percolation and harmonic Dirichlet functions. Geom. Funct. Anal. 15(5), 1004–1051. MR 222115710.1007/s00039-005-0539-2CrossRefGoogle Scholar
Gaboriau, D (2010) Orbit equivalence and measured group theory. In Proceedings of the International Congress of Mathematicians vol. III. New Delhi: Hindustan Book Agency, 1501–1527. MR 2827853Google Scholar
Ge, L (1998) Applications of free entropy to finite von Neumann algebras. II. Ann. of Math. (2) 147(1), 143–157. MR 160952210.2307/120985CrossRefGoogle Scholar
Gangbo, W, Jekel, D, Nam, K and Shlyakhtenko, D (2022) Duality for optimal couplings in free probability. Communications in Mathematical Physics 396(3), 903–981.10.1007/s00220-022-04480-0CrossRefGoogle Scholar
Ge, L and Shen, J (2002) On free entropy dimension of finite von Neumann algebras. Geom. Funct. Anal. 12(3), 546–566. MR 192437110.1007/s00039-002-8256-6CrossRefGoogle Scholar
Guichardet, A (1972) Sur la cohomologie des groupes topologiques. II. Bull. Sci. Math. (2)
96, 305–332. MR 340464Google Scholar
Glasner, E and Weiss, B (1997) Kazhdan’s property T and the geometry of the collection of invariant measures. Geom. Funct. Anal. 7(5), 917–935. MR 147555010.1007/s000390050030CrossRefGoogle Scholar
Hayes, B (2018) 1-bounded entropy and regularity problems in von Neumann algebras. Int. Math. Res. Not. IMRN (1), 57–137. MR 3801429Google Scholar
Hayes, B, Jekel, D, Nelson, B and Sinclair, T (2021) A random matrix approach to absorption in free products. Int. Math. Res. Not. IMRN (3), 1919–1979. MR 420660110.1093/imrn/rnaa191CrossRefGoogle Scholar
Houdayer, C and Shlyakhtenko, D (2011) Strongly solid
$\mathrm{I}{\mathrm{I}}_1$
factors with an exotic MASA. Int. Math. Res. Not. IMRN (6), 1352–1380. MR 2806507Google Scholar
$\mathrm{I}{\mathrm{I}}_1$
factors with an exotic MASA. Int. Math. Res. Not. IMRN (6), 1352–1380. MR 2806507Google ScholarIoana, A (2011) Cocycle superrigidity for profinite actions of property (T) groups. Duke Math. J. 157(2), 337–367. MR 278393310.1215/00127094-2011-008CrossRefGoogle Scholar
Ioana, A (2011)
${W}^{\ast }$
-superrigidity for Bernoulli actions of property (T) groups. J. Amer. Math. Soc. 24(4), 1175–1226. MR 281334110.1090/S0894-0347-2011-00706-6CrossRefGoogle Scholar
${W}^{\ast }$
-superrigidity for Bernoulli actions of property (T) groups. J. Amer. Math. Soc. 24(4), 1175–1226. MR 281334110.1090/S0894-0347-2011-00706-6CrossRefGoogle ScholarJones, VFR (1983) Index for subfactors. Invent. Math. 72(1), 1–25. MR 69668810.1007/BF01389127CrossRefGoogle Scholar
Jung, K and Shlyakhtenko, D (2007) Any generating set of an arbitrary property T von Neumann algebra has free entropy dimension
$\le 1$
. J. Noncommut. Geom. 1(2), 271–279. MR 230830710.4171/jncg/7CrossRefGoogle Scholar
$\le 1$
. J. Noncommut. Geom. 1(2), 271–279. MR 230830710.4171/jncg/7CrossRefGoogle ScholarJung, K The rank theorem and
${L}^2$
-invariants in free entropy: Global upper bounds. Preprint, 2016, arXiv:1602.04726.Google Scholar
${L}^2$
-invariants in free entropy: Global upper bounds. Preprint, 2016, arXiv:1602.04726.Google ScholarJung, K Some free entropy dimension inequalities for subfactors. Preprint, 2004, arXiv:0410594.Google Scholar
Jung, K (2003) A free entropy dimension lemma. Pacific J. Math. 211(2), 265–271. MR 201573610.2140/pjm.2003.211.265CrossRefGoogle Scholar
Jung, K (2003) The free entropy dimension of hyperfinite von Neumann algebras. Trans. Amer. Math. Soc. 355(12), 5053–5089. MR 199759510.1090/S0002-9947-03-03286-0CrossRefGoogle Scholar
Jung, K (2006) A hyperfinite inequality for free entropy dimension. Proc. Amer. Math. Soc. 134(7), 2099–2108. MR 221578010.1090/S0002-9939-06-08237-2CrossRefGoogle Scholar
Jung, K (2007) Amenability, tubularity, and embeddings into
${\mathcal{R}}^{\omega }$
. Math. Ann. 338(1), 241–248. MR 229551110.1007/s00208-006-0074-yCrossRefGoogle Scholar
${\mathcal{R}}^{\omega }$
. Math. Ann. 338(1), 241–248. MR 229551110.1007/s00208-006-0074-yCrossRefGoogle ScholarJung, K (2007) Strongly
$1$
-bounded von Neumann algebras. Geom. Funct. Anal. 17(4), 1180–1200. MR 237301410.1007/s00039-007-0624-9CrossRefGoogle Scholar
$1$
-bounded von Neumann algebras. Geom. Funct. Anal. 17(4), 1180–1200. MR 237301410.1007/s00039-007-0624-9CrossRefGoogle ScholarKaždan, DA (1967) On the connection of the dual space of a group with the structure of its closed subgroups. Funkcional. Anal. i Priložen. 1, 71–74. MR 0209390Google Scholar
Lyons, R and Schramm, O (1999) Indistinguishability of percolation clusters. Ann. Probab. 27(4), 1809–1836. MR 174288910.1214/aop/1022874816CrossRefGoogle Scholar
Margulis, GA (1973) Explicit constructions of expanders. Problemy Peredači Informacii 9(4), 71–80. MR 0484767Google Scholar
Jung, K (1979) Finiteness of quotient groups of discrete subgroups. Funktsional. Anal. i Prilozhen. 13(3), 28–39. MR 545365Google Scholar
Ozawa, N and Popa, S (2010) On a class of
$\mathrm{I}{\mathrm{I}}_1$
factors with at most one Cartan subalgebra. Ann. of Math. (2) 172(1), 713–749. MR 268043010.4007/annals.2010.172.713CrossRefGoogle Scholar
$\mathrm{I}{\mathrm{I}}_1$
factors with at most one Cartan subalgebra. Ann. of Math. (2) 172(1), 713–749. MR 268043010.4007/annals.2010.172.713CrossRefGoogle ScholarOzawa, N and Popa, S (2010) On a class of
$\mathrm{I}{\mathrm{I}}_1$
factors with at most one Cartan subalgebra, II. Amer. J. Math. 132(3), 841–866. MR 266690910.1353/ajm.0.0121CrossRefGoogle Scholar
$\mathrm{I}{\mathrm{I}}_1$
factors with at most one Cartan subalgebra, II. Amer. J. Math. 132(3), 841–866. MR 266690910.1353/ajm.0.0121CrossRefGoogle ScholarOzawa, N (2004) Solid von Neumann algebras. Acta Math. 192(1), 111–117. MR 207960010.1007/BF02441087CrossRefGoogle Scholar
Ozawa, N (2004) There is no separable universal
$\mathrm{I}{\mathrm{I}}_1$
-factor. Proc. Amer. Math. Soc. 132(2), 487–490. MR 202237310.1090/S0002-9939-03-07127-2CrossRefGoogle Scholar
$\mathrm{I}{\mathrm{I}}_1$
-factor. Proc. Amer. Math. Soc. 132(2), 487–490. MR 202237310.1090/S0002-9939-03-07127-2CrossRefGoogle ScholarPeterson, J (2009) A
$1$
-cohomology characterization of property (t) in von neumann algebras. Pacific Journal of Math. 243(1), 181–199.10.2140/pjm.2009.243.181CrossRefGoogle Scholar
$1$
-cohomology characterization of property (t) in von neumann algebras. Pacific Journal of Math. 243(1), 181–199.10.2140/pjm.2009.243.181CrossRefGoogle ScholarPeterson, J (2009)
${L}^2$
-rigidity in von Neumann algebras. Invent. Math. 175(2), 417–433. MR 247011110.1007/s00222-008-0154-6CrossRefGoogle Scholar
${L}^2$
-rigidity in von Neumann algebras. Invent. Math. 175(2), 417–433. MR 247011110.1007/s00222-008-0154-6CrossRefGoogle ScholarPopa, S (2004) On the fundamental group of type
$\mathrm{I}{\mathrm{I}}_1$
factors. Proc. Natl. Acad. Sci. USA 101(3), 723–726. MR 202917710.1073/pnas.2333960100CrossRefGoogle Scholar
$\mathrm{I}{\mathrm{I}}_1$
factors. Proc. Natl. Acad. Sci. USA 101(3), 723–726. MR 202917710.1073/pnas.2333960100CrossRefGoogle ScholarPopa, S (2006) On a class of type
$\mathrm{I}{\mathrm{I}}_1$
factors with Betti numbers invariants. Ann. of Math. (2) 163(3), 809–899. MR 221513510.4007/annals.2006.163.809CrossRefGoogle Scholar
$\mathrm{I}{\mathrm{I}}_1$
factors with Betti numbers invariants. Ann. of Math. (2) 163(3), 809–899. MR 221513510.4007/annals.2006.163.809CrossRefGoogle ScholarPopa, S (2006) Strong rigidity of
$\mathrm{II}_1$
factors arising from malleable actions of
$w$
-rigid groups. I. Invent. Math. 165(2), 369–408. MR 223196110.1007/s00222-006-0501-4CrossRefGoogle Scholar
$\mathrm{II}_1$
factors arising from malleable actions of
$w$
-rigid groups. I. Invent. Math. 165(2), 369–408. MR 223196110.1007/s00222-006-0501-4CrossRefGoogle ScholarPopa, S (2006) Strong rigidity of
$I{I}_1$
factors arising from malleable actions of
$w$
-rigid groups. II. Invent. Math. 165(2), 409–451. MR 223196210.1007/s00222-006-0502-3CrossRefGoogle Scholar
$I{I}_1$
factors arising from malleable actions of
$w$
-rigid groups. II. Invent. Math. 165(2), 409–451. MR 223196210.1007/s00222-006-0502-3CrossRefGoogle ScholarPopa, S (2007) Cocycle and orbit equivalence superrigidity for malleable actions of
$w$
-rigid groups. Invent. Math. 170(2), 243–295. MR 234263710.1007/s00222-007-0063-0CrossRefGoogle Scholar
$w$
-rigid groups. Invent. Math. 170(2), 243–295. MR 234263710.1007/s00222-007-0063-0CrossRefGoogle ScholarPopa, S (2014) Independence properties in subalgebras of ultraproduct
$\mathrm{I}{\mathrm{I}}_1$
factors. J. Funct. Anal. 266(9), 5818–5846. MR 318296110.1016/j.jfa.2014.02.004CrossRefGoogle Scholar
$\mathrm{I}{\mathrm{I}}_1$
factors. J. Funct. Anal. 266(9), 5818–5846. MR 318296110.1016/j.jfa.2014.02.004CrossRefGoogle ScholarPeterson, J and Popa, S (2005) On the notion of relative property (T) for inclusions of von Neumann algebras. J. Funct. Anal. 219(2), 469–483. MR 210926010.1016/j.jfa.2004.04.017CrossRefGoogle Scholar
Popa, S and Vaes, S (2014) Unique Cartan decomposition for
$\mathrm{I}{\mathrm{I}}_1$
factors arising from arbitrary actions of free groups. Acta Math. 212 (1), 141–198. MR 317960910.1007/s11511-014-0110-9CrossRefGoogle Scholar
$\mathrm{I}{\mathrm{I}}_1$
factors arising from arbitrary actions of free groups. Acta Math. 212 (1), 141–198. MR 317960910.1007/s11511-014-0110-9CrossRefGoogle ScholarPopa, S and Vaes, S (2014) Unique Cartan decomposition for
$I{I}_1$
factors arising from arbitrary actions of hyperbolic groups. J. Reine Angew. Math. 694, 215–239. MR 325904410.1515/crelle-2012-0104CrossRefGoogle Scholar
$I{I}_1$
factors arising from arbitrary actions of hyperbolic groups. J. Reine Angew. Math. 694, 215–239. MR 325904410.1515/crelle-2012-0104CrossRefGoogle ScholarRădulescu, F (1994) Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index. Invent. Math. 115(2), 347–389. MR 125890910.1007/BF01231764CrossRefGoogle Scholar
Reed, M and Simon, B (1980) Methods of Modern Mathematical Physics. I, 2nd edn. New York: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]. MR 751959Google Scholar
Schmidt, K (1980) Asymptotically invariant sequences and an action of SL(2,Z) on the
$2$
-sphere. Israel J. Math. 37(3), 193–208. MR 59945410.1007/BF02760961CrossRefGoogle Scholar
$2$
-sphere. Israel J. Math. 37(3), 193–208. MR 59945410.1007/BF02760961CrossRefGoogle ScholarSchmidt, K (1981) Amenability, Kazhdan’s property
$T$
, strong ergodicity and invariant means for ergodic group-actions. Ergodic Theory Dynamical Systems 1(2), 223–236. MR 661821Google Scholar
$T$
, strong ergodicity and invariant means for ergodic group-actions. Ergodic Theory Dynamical Systems 1(2), 223–236. MR 661821Google ScholarShlyakhtenko, D (2021) Von Neumann algebras of sofic groups with
${\beta}_1^{(2)}=0$
are strongly 1-bounded. J. Operator Theory 85 (1), 217–228.10.7900/jot.2019oct21.2270CrossRefGoogle Scholar
${\beta}_1^{(2)}=0$
are strongly 1-bounded. J. Operator Theory 85 (1), 217–228.10.7900/jot.2019oct21.2270CrossRefGoogle ScholarSzarek, SJ (1998) Metric entropy of homogeneous spaces. In Quantum Probability (Gdańsk, 1997), Banach Center Publ., vol. 43. Warsaw: Polish Acad. Sci. Inst. Math., 395–410. MR 1649741Google Scholar
Voiculescu, D (1990) Circular and semicircular systems and free product factors. In Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Paris, 1989), Progr. Math., vol. 92. Boston: Birkhäuser Boston, 45–60. MR 1103585Google Scholar
Voiculescu, DV (1994) The analogues of entropy and of Fisher’s information measure in free probability theory II. Invent. Math. 118 (3), 411–440. MR 129635210.1007/BF01231539CrossRefGoogle Scholar
Voiculescu, DV (1996) The analogues of entropy and of Fisher’s information measure in free probability theory. III. The absence of Cartan subalgebras. Geom. Funct. Anal. 6(1), 172–199. MR 137123610.1007/BF02246772CrossRefGoogle Scholar
Voiculescu, DV (1998) A strengthened asymptotic freeness result for random matrices with applications to free entropy. Internat. Math. Res. Not. IMRN (1), 41–63. MR 160187810.1155/S107379289800004XCrossRefGoogle Scholar
Voiculescu, D (1999) Free entropy dimension
$\le 1$
for some generators of property
$\mathrm{T}$
factors of type
$I{I}_1$
J. Reine Angew. Math. 514, 113–118. MR 171128310.1515/crll.1999.067CrossRefGoogle Scholar
$\le 1$
for some generators of property
$\mathrm{T}$
factors of type
$I{I}_1$
J. Reine Angew. Math. 514, 113–118. MR 171128310.1515/crll.1999.067CrossRefGoogle Scholar