## Kyoto Journal of Mathematics

### Examples of groups which are not weakly amenable

Narutaka Ozawa

#### Abstract

We prove that weak amenability of a locally compact group imposes a strong condition on its amenable closed normal subgroups. This extends non–weak amenability results of Haagerup (1988) and Ozawa and Popa (2010). A von Neumann algebra analogue is also obtained.

#### Article information

Source
Kyoto J. Math. Volume 52, Number 2 (2012), 333-344.

Dates
First available in Project Euclid: 24 April 2012

http://projecteuclid.org/euclid.kjm/1335272982

Digital Object Identifier
doi:10.1215/21562261-1550985

Mathematical Reviews number (MathSciNet)
MR2914879

Zentralblatt MATH identifier
1242.43007

#### Citation

Ozawa, Narutaka. Examples of groups which are not weakly amenable. Kyoto J. Math. 52 (2012), no. 2, 333--344. doi:10.1215/21562261-1550985. http://projecteuclid.org/euclid.kjm/1335272982.

#### References

• [BO] N. Brown and N. Ozawa, C*-algebras and Finite-Dimensional Approximations, Grad. Studies in Math. 88, Amer. Math. Soc., Providence, 2008.
• [dCH] J. de Cannière and U. Haagerup, Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. J. Math. 107 (1985), 455–500.
• [Co] M. Cowling, “Harmonic analysis on some nilpotent Lie groups (with application to the representation theory of some semisimple Lie groups)” in Topics in Modern Harmonic Analysis, Vols. I, II (Turin/Milan, 1982), Ist. Naz. Alta Mat. Francesco Severi, Rome, 1983, 81–123.
• [CH] M. Cowling and U. Haagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math. 96 (1989), 507–549.
• [CZ] M. Cowling and R. J. Zimmer, Actions of lattices in Sp(1, n), Ergodic Theory Dynam. Systems 9 (1989), 221–237.
• [Do] B. Dorofaeff, The Fourier algebra of SL(2, R) ⋊ Rn, n ≥ 2, has no multiplier bounded approximate unit, Math. Ann. 297 (1993), 707–724.
• [Ha] U. Haagerup, Group C*-algebras without the completely bounded approximation property, preprint, 1988.
• [HK] U. Haagerup and J. Kraus, Approximation properties for group C*-algebras and group von Neumann algebras, Trans. Amer. Math. Soc. 344 (1994), 667–699.
• [Jo] P. Jolissaint, A characterization of completely bounded multipliers of Fourier algebras, Colloq. Math. 63 (1992), 311–313.
• [LdS] V. Lafforgue and M. de la Salle, Noncommutative Lp-spaces without the completely bounded approximation property, Duke Math. J. 160 (2011), 71–116.
• [Mo] N. Monod, “An invitation to bounded cohomology” in International Congress of Mathematicians, Vol. II, Eur. Math. Soc., Zürich, 2006, 1183–1211.
• [Oz] N. Ozawa, Weak amenability of hyperbolic groups, Groups Geom. Dyn. 2 (2008), 271–280.
• [OP] N. Ozawa and S. Popa, On a class of II1 factors with at most one Cartan subalgbra, Ann. of Math. (2) 172 (2010), 713–749.
• [Sa] H. Sako, The class $\mathcal{S}$ as an ME invariant, Int. Math. Res. Not. IMRN 2009, no. 15, 2749–2759.
• [Ta] M. Takesaki, Theory of operator algebras, I, reprint of the first (1979) ed., Encyclopaedia Math. Sci. 124, Operator Algebras and Non-commutative Geometry 5, Springer, Berlin, 2002.