## Kyoto Journal of Mathematics

### Weak amenability and simply connected Lie groups

Søren Knudby

#### Abstract

Following an approach of Ozawa, we show that several semidirect products are not weakly amenable. As a consequence, we are able to completely characterize the simply connected Lie groups that are weakly amenable.

#### Article information

Source
Kyoto J. Math., Volume 56, Number 3 (2016), 689-700.

Dates
Revised: 6 July 2015
Accepted: 8 July 2015
First available in Project Euclid: 22 August 2016

https://projecteuclid.org/euclid.kjm/1471872288

Digital Object Identifier
doi:10.1215/21562261-3600238

Mathematical Reviews number (MathSciNet)
MR3542782

Zentralblatt MATH identifier
1348.43004

#### Citation

Knudby, Søren. Weak amenability and simply connected Lie groups. Kyoto J. Math. 56 (2016), no. 3, 689--700. doi:10.1215/21562261-3600238. https://projecteuclid.org/euclid.kjm/1471872288

#### References

• [1] B. Bekka, P. de la Harpe, and A. Valette, Kazhdan’s Property (T), New Math. Monogr. 11, Cambridge Univ. Press, Cambridge, 2008.
• [2] M. Bożejko and G. Fendler, Herz–Schur multipliers and completely bounded multipliers of the Fourier algebra of a locally compact group, Boll. Un. Mat. Ital. A (6) 3 (1984), 297–302.
• [3] N. P. Brown and N. Ozawa, $C^{*}$-Algebras and Finite-Dimensional Approximations, Grad. Stud. Math. 88, Amer. Math. Soc., Providence, 2008.
• [4] M. Cowling, Rigidity for lattices in semisimple Lie groups: von Neumann algebras and ergodic actions, Rend. Sem. Mat. Univ. Politec. Torino 47 (1989), 1–37.
• [5] M. Cowling, B. Dorofaeff, A. Seeger, and J. Wright, A family of singular oscillatory integral operators and failure of weak amenability, Duke Math. J. 127 (2005), 429–486.
• [6] 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.
• [7] Y. de Cornulier, Kazhdan and Haagerup properties in algebraic groups over local fields, J. Lie Theory 16 (2006), 67–82.
• [8] B. Dorofaeff, The Fourier algebra of $\mathrm{SL}(2,\mathbf{R})\rtimes\mathbf{R}^{n}$, $n\geq2$, has no multiplier bounded approximate unit, Math. Ann. 297 (1993), 707–724.
• [9] B. Dorofaeff, Weak amenability and semidirect products in simple Lie groups, Math. Ann. 306 (1996), 737–742.
• [10] E. Guentner and N. Higson, Weak amenability of $\mathrm{CAT}(0)$-cubical groups, Geom. Dedicata 148 (2010), 137–156.
• [11] U. Haagerup, Group $C^{*}$-algebras without the completely bounded approximation property, unpublished manuscript, 1986.
• [12] U. Haagerup and T. de Laat, Simple Lie groups without the approximation property, Duke Math. J. 162 (2013), 925–964.
• [13] U. Haagerup and T. de Laat, Simple Lie groups without the approximation property, II, Trans. Amer. Math. Soc. 368 (2016), no. 6, 3777–3809.
• [14] U. Haagerup, S. Knudby, and T. de Laat, A complete characterization of connected Lie groups with the approximation property, to appear in Ann. Sci. Ec. Norm. Sup. 49 (2016), preprint, arXiv:1412.3033v2 [math.GR].
• [15] M. L. Hansen, Weak amenability of the universal covering group of $\mathrm{SU}(1,n)$, Math. Ann. 288 (1990), 445–472.
• [16] P. Jolissaint, A characterization of completely bounded multipliers of Fourier algebras, Colloq. Math. 63 (1992), 311–313.
• [17] P. Jolissaint, Proper cocycles and weak forms of amenability, Colloq. Math. 138 (2015), 73–88.
• [18] V. Lafforgue, Un analogue non archimédien d’un résultat de haagerup et lien avec la propriété (t) renforcée, to appear in J. Noncommut. Geom. (2016).
• [19] S. Lang, $\mathrm{SL}_{2}(\mathbf{R})$, Addison-Wesley, Reading, Mass., 1975.
• [20] G. D. Mostow, The extensibility of local Lie groups of transformations and groups on surfaces, Ann. of Math. (2) 52 (1950), 606–636.
• [21] N. Ozawa, Weak amenability of hyperbolic groups, Groups Geom. Dyn. 2 (2008), 271–280.
• [22] N. Ozawa, Examples of groups which are not weakly amenable, Kyoto J. Math. 52 (2012), 333–344.
• [23] N. Ozawa and S. Popa, On a class of $\mathrm{II}_{1}$ factors with at most one Cartan subalgebra, Ann. of Math. (2) 172 (2010), 713–749.
• [24] G. Pisier, Similarity Problems and Completely Bounded Maps, Lecture Notes in Math. 1618, Springer, Berlin, 2001.
• [25] V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Pure Appl. Math. 139, Academic Press, Boston, 1994.
• [26] V. S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations, reprint of the 1974 ed., Grad. Texts in Math. 102, Springer, New York, 1984.
• [27] H. Whitney, Elementary structure of real algebraic varieties, Ann. of Math. (2) 66 (1957), 545–556.