Acta Mathematica

An inner amenable group whose von Neumann algebra does not have property Gamma

Stefaan Vaes

Full-text: Open access


We construct inner amenable groups G with infinite conjugacy classes and such that the associated II1 factor has no non-trivial asymptotically central elements, i.e. does not have property Gamma of Murray and von Neumann. This solves a problem posed by Effros in 1975.

Article information

Acta Math., Volume 208, Number 2 (2012), 389-394.

Received: 18 March 2010
Revised: 15 July 2011
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2012 © Institut Mittag-Leffler


Vaes, Stefaan. An inner amenable group whose von Neumann algebra does not have property Gamma. Acta Math. 208 (2012), no. 2, 389--394. doi:10.1007/s11511-012-0079-1.

Export citation


  • B édos, E. & de la H arpe, P., Moyennabilité intérieure des groupes: définitions et exemples. Enseign. Math., 32 (1986), 139–157.
  • B ekka, B., de la H arpe, P. & V alette, A., Kazhdan’s Property (T). New Mathematical Monographs, 11. Cambridge University Press, Cambridge, 2008.
  • C onnes, A., Classification of injective factors. Cases II1, II, IIIλ, λ≠1. Ann. of Math., 104 (1976), 73–115.
  • E ffros, E. G., Property Γ and inner amenability. Proc. Amer. Math. Soc., 47 (1975), 483–486.
  • de la H arpe, P., Operator algebras, free groups and other groups, in Recent Advances in OperatorAlgebras (Orléans, 1992). Astérisque, 232 (1995), 121–153.
  • J olissaint, P., Moyennabilité intérieure du groupe F de Thompson. C. R. Acad. Sci.Paris Sér. I Math., 325 (1997), 61–64.
  • — Central sequences in the factor associated with Thompson’s group F. Ann. Inst. Fourier (Grenoble), 48 (1998), 1093–1106.
  • M urray, F. J. & von N eumann, J., On rings of operators IV. Ann. of Math., 44 (1943), 716–808.
  • S chmidt, K., Amenability, Kazhdan’s property T, strong ergodicity and invariant means for ergodic group-actions. Ergodic Theory Dynam. Systems, 1 (1981), 223–236.
  • S talder, Y., Moyennabilité intérieure et extensions HNN. Ann. Inst. Fourier (Grenoble), 56 (2006), 309–323.