The weak Haagerup property

Knudby, Søren

doi:10.1090/tran/6445

The weak Haagerup property
HTML articles powered by AMS MathViewer

by Søren Knudby PDF

Trans. Amer. Math. Soc. 368 (2016), 3469-3508 Request permission

Abstract:

We introduce the weak Haagerup property for locally compact groups and prove several hereditary results for the class of groups with this approximation property. The class contains a priori all weakly amenable groups and groups with the usual Haagerup property, but examples are given of groups with the weak Haagerup property which are not weakly amenable and do not have the Haagerup property.

In the second part of the paper we introduce the weak Haagerup property for finite von Neumann algebras, and we prove several hereditary results here as well. Also, a discrete group has the weak Haagerup property if and only if its group von Neumann algebra does.

Finally, we give an example of two $\mathrm {II}_1$ factors with different weak Haagerup constants.

References

C. Anantharaman-Delaroche, Amenable correspondences and approximation properties for von Neumann algebras, Pacific J. Math. 171 (1995), no. 2, 309–341. MR 1372231
Claire Anantharaman-Delaroche, On ergodic theorems for free group actions on noncommutative spaces, Probab. Theory Related Fields 135 (2006), no. 4, 520–546. MR 2240699, DOI 10.1007/s00440-005-0456-1
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
Armand Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485–535. MR 147566, DOI 10.2307/1970210
Marek Bożejko and Gero 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), no. 2, 297–302 (English, with Italian summary). MR 753889
Michael Brannan and Brian Forrest, Extending multipliers of the Fourier algebra from a subgroup, Proc. Amer. Math. Soc. 142 (2014), no. 4, 1181–1191. MR 3162241, DOI 10.1090/S0002-9939-2014-11824-7
Nathanial P. Brown and Narutaka Ozawa, $C^*$-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008. MR 2391387, DOI 10.1090/gsm/088
Pierre-Alain Cherix, Michael Cowling, Paul Jolissaint, Pierre Julg, and Alain Valette, Groups with the Haagerup property, Progress in Mathematics, vol. 197, Birkhäuser Verlag, Basel, 2001. Gromov’s a-T-menability. MR 1852148, DOI 10.1007/978-3-0348-8237-8
Hisashi Choda, An extremal property of the polar decomposition in von Neumann algebras, Proc. Japan Acad. 46 (1970), 341–344. MR 355624
Marie Choda, Group factors of the Haagerup type, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 5, 174–177. MR 718798
Man Duen Choi and Edward G. Effros, Separable nuclear $C^*$-algebras and injectivity, Duke Math. J. 43 (1976), no. 2, 309–322. MR 405117
Man Duen Choi and Edward G. Effros, Nuclear $C^*$-algebras and injectivity: the general case, Indiana Univ. Math. J. 26 (1977), no. 3, 443–446. MR 430794, DOI 10.1512/iumj.1977.26.26034
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
Michael Cowling, Harmonic analysis on some nilpotent Lie groups (with application to the representation theory of some semisimple Lie groups), Topics in modern harmonic analysis, Vol. I, II (Turin/Milan, 1982) Ist. Naz. Alta Mat. Francesco Severi, Rome, 1983, pp. 81–123. MR 748862
Michael Cowling, Brian Dorofaeff, Andreas Seeger, and James Wright, A family of singular oscillatory integral operators and failure of weak amenability, Duke Math. J. 127 (2005), no. 3, 429–486. MR 2132866, DOI 10.1215/S0012-7094-04-12732-0
Michael Cowling and Uffe Haagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math. 96 (1989), no. 3, 507–549. MR 996553, DOI 10.1007/BF01393695
Jean De Cannière and Uffe Haagerup, Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. J. Math. 107 (1985), no. 2, 455–500. MR 784292, DOI 10.2307/2374423
Yves de Cornulier, Yves Stalder, and Alain Valette, Proper actions of lamplighter groups associated with free groups, C. R. Math. Acad. Sci. Paris 346 (2008), no. 3-4, 173–176 (English, with English and French summaries). MR 2393636, DOI 10.1016/j.crma.2007.11.027
Brian Dorofaeff, The Fourier algebra of $\textrm {SL}(2,\textbf {R})\rtimes \textbf {R}^n$, $n\geq 2$, has no multiplier bounded approximate unit, Math. Ann. 297 (1993), no. 4, 707–724. MR 1245415, DOI 10.1007/BF01459526
Brian Dorofaeff, Weak amenability and semidirect products in simple Lie groups, Math. Ann. 306 (1996), no. 4, 737–742. MR 1418350, DOI 10.1007/BF01445274
Edward G. Effros and Zhong-Jin Ruan, Operator spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000. MR 1793753
Pierre Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236 (French). MR 228628
Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397028
Gerald B. Folland, Real analysis, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999. Modern techniques and their applications; A Wiley-Interscience Publication. MR 1681462
Uffe Haagerup, An example of a nonnuclear $C^{\ast }$-algebra, which has the metric approximation property, Invent. Math. 50 (1978/79), no. 3, 279–293. MR 520930, DOI 10.1007/BF01410082
Uffe Haagerup, Group $C^*$-algebras without the completely bounded approximation property, unpublished manuscript, 1986.
Uffe Haagerup and Tim de Laat, Simple Lie groups without the approximation property, Duke Math. J. 162 (2013), no. 5, 925–964. MR 3047470, DOI 10.1215/00127094-2087672
Uffe Haagerup and Tim de Laat, Simple Lie groups without the approximation property, II, Trans. Amer. Math. Soc., to appear.
Uffe Haagerup and Søren Knudby, The weak Haagerup property II: Examples, Int. Math. Res. Notices, first published online September 20, 2014, DOI 10.1093/imrn/rnu132.
Uffe Haagerup and Jon Kraus, Approximation properties for group $C^*$-algebras and group von Neumann algebras, Trans. Amer. Math. Soc. 344 (1994), no. 2, 667–699. MR 1220905, DOI 10.1090/S0002-9947-1994-1220905-3
Mogens Lemvig Hansen, Weak amenability of the universal covering group of $\textrm {SU}(1,n)$, Math. Ann. 288 (1990), no. 3, 445–472. MR 1079871, DOI 10.1007/BF01444541
Nigel Higson and Gennadi Kasparov, Operator $K$-theory for groups which act properly and isometrically on Hilbert space, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 131–142. MR 1487204, DOI 10.1090/S1079-6762-97-00038-3
Nigel Higson and Gennadi Kasparov, $E$-theory and $KK$-theory for groups which act properly and isometrically on Hilbert space, Invent. Math. 144 (2001), no. 1, 23–74. MR 1821144, DOI 10.1007/s002220000118
Paul Jolissaint, A characterization of completely bounded multipliers of Fourier algebras, Colloq. Math. 63 (1992), no. 2, 311–313. MR 1180643, DOI 10.4064/cm-63-2-311-313
Paul Jolissaint, Borel cocycles, approximation properties and relative property T, Ergodic Theory Dynam. Systems 20 (2000), no. 2, 483–499. MR 1756981, DOI 10.1017/S0143385700000237
Paul Jolissaint, Haagerup approximation property for finite von Neumann algebras, J. Operator Theory 48 (2002), no. 3, suppl., 549–571. MR 1962471
Paul Jolissaint, Proper cocycles and weak forms of amenability, Colloq. Math. 138 (2015), no. 1, 73–88. MR 3310701, DOI 10.4064/cm138-1-5
Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. II, Graduate Studies in Mathematics, vol. 16, American Mathematical Society, Providence, RI, 1997. Advanced theory; Corrected reprint of the 1986 original. MR 1468230, DOI 10.1090/gsm/016/01
Søren Knudby, Semigroups of Herz-Schur multipliers, J. Funct. Anal. 266 (2014), no. 3, 1565–1610. MR 3146826, DOI 10.1016/j.jfa.2013.11.002
Bertram Kostant, On the existence and irreducibility of certain series of representations, Bull. Amer. Math. Soc. 75 (1969), 627–642. MR 245725, DOI 10.1090/S0002-9904-1969-12235-4
Bertram Kostant, On the existence and irreducibility of certain series of representations, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) Halsted, New York, 1975, pp. 231–329. MR 0399361
Vincent Lafforgue and Mikael De la Salle, Noncommutative $L^p$-spaces without the completely bounded approximation property, Duke Math. J. 160 (2011), no. 1, 71–116. MR 2838352, DOI 10.1215/00127094-1443478
Horst Leptin, Sur l’algèbre de Fourier d’un groupe localement compact, C. R. Acad. Sci. Paris Sér. A-B 266 (1968), A1180–A1182 (French). MR 239002
George W. Mackey, Induced representations of locally compact groups. I, Ann. of Math. (2) 55 (1952), 101–139. MR 44536, DOI 10.2307/1969423
Narutaka Ozawa, Examples of groups which are not weakly amenable, Kyoto J. Math. 52 (2012), no. 2, 333–344. MR 2914879, DOI 10.1215/21562261-1550985
Narutaka Ozawa and Sorin Popa, On a class of $\textrm {II}_1$ factors with at most one Cartan subalgebra, Ann. of Math. (2) 172 (2010), no. 1, 713–749. MR 2680430, DOI 10.4007/annals.2010.172.713
Gert K. Pedersen and Masamichi Takesaki, The Radon-Nikodym theorem for von Neumann algebras, Acta Math. 130 (1973), 53–87. MR 412827, DOI 10.1007/BF02392262
Jean-Paul Pier, Amenable locally compact groups, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 767264
Gilles Pisier, Similarity problems and completely bounded maps, Second, expanded edition, Lecture Notes in Mathematics, vol. 1618, Springer-Verlag, Berlin, 2001. Includes the solution to “The Halmos problem”. MR 1818047, DOI 10.1007/b55674
Jean-Philippe Préaux, Group extensions with infinite conjugacy classes, Confluentes Math. 5 (2013), no. 1, 73–92. MR 3143612, DOI 10.5802/cml.3
Éric Ricard and Quanhua Xu, Khintchine type inequalities for reduced free products and applications, J. Reine Angew. Math. 599 (2006), 27–59. MR 2279097, DOI 10.1515/CRELLE.2006.077
Allan M. Sinclair and Roger R. Smith, The Haagerup invariant for von Neumann algebras, Amer. J. Math. 117 (1995), no. 2, 441–456. MR 1323684, DOI 10.2307/2374923
Allan M. Sinclair and Roger R. Smith, Finite von Neumann algebras and masas, London Mathematical Society Lecture Note Series, vol. 351, Cambridge University Press, Cambridge, 2008. MR 2433341, DOI 10.1017/CBO9780511666230
Troels Steenstrup, Fourier multiplier norms of spherical functions on the generalized Lorentz groups, preprint, arXiv:0911.4977, 2009.
M. Takesaki, Theory of operator algebras. I, Encyclopaedia of Mathematical Sciences, vol. 124, Springer-Verlag, Berlin, 2002. Reprint of the first (1979) edition; Operator Algebras and Non-commutative Geometry, 5. MR 1873025
John von Neumann, Zur allgemeinen Theorie des Maßes, Fund. Math. 13 (1929), 73–116.
Simon Wassermann, Injective $W^*$-algebras, Math. Proc. Cambridge Philos. Soc. 82 (1977), no. 1, 39–47. MR 448108, DOI 10.1017/S0305004100053664
Guangwu Xu, Herz-Schur multipliers and weakly almost periodic functions on locally compact groups, Trans. Amer. Math. Soc. 349 (1997), no. 6, 2525–2536. MR 1373647, DOI 10.1090/S0002-9947-97-01733-9