Noncommutative \(L^{p}\)-spaces without the completely bounded approximation property. (English) Zbl 1267.46072

Lafforgue, Vincent; de la Salle, Mikael

Duke Math. J. 160, No. 1, 71-116 (2011).

Summary: For any \(1\leq p\leq\infty\) different from 2, we give examples of noncommutative \(L^{p}\)-spaces without the completely bounded approximation property. Let \(F\) be a non-Archimedian local field. If \(p > 4\) or \(p < 4/3\) and \(r\geq 3\), these examples are the noncommutative \(L^{p}\)-spaces of the von Neumann algebra of lattices in SL\(_{r}(F)\) or in SL\(_{r}({\mathbb{R}})\). For other values of \(p\), the examples are the noncommutative \(L^{p}\)-spaces of the von Neumann algebra of lattices in SL\(_{r}(F)\) for \(r\) large enough depending on \(p\).
We also prove that, if \(r\geq3\), lattices in SL\(_{r}(F)\) or SL\(_{r}({\mathbb{R}})\) do not have the approximation property of Haagerup and Kraus. This provides examples of exact \(C^{*}\)-algebras without the operator space approximation property.

MSC:

46L07	Operator spaces and completely bounded maps
22D25	\(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
46B28	Spaces of operators; tensor products; approximation properties

Keywords:

noncommutative \(L^{p}\)-spaces; completely bounded approximation property; exact \(C^{*}\)-algebras; operator space approximation property

References:

[1]	N. Bourbaki, Éléments de mathématique, Fasc. XIII, Livre VI: Intégration , 2nd ed., Actualités Scientifiques et Industrielles 1175 , Hermann, Paris, 1965.
[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. · Zbl 0564.43004
[3]	M. Bożejko and G. Fendler, Herz-Schur multipliers and uniformly bounded representations of discrete groups , Arch. Math. 57 (1991), 290-298. · Zbl 0726.43007
[4]	N. P. Brown and N. Ozawa, C \ast - Algebras and Finite-Dimensional Approximations , Grad. Stud. Math. 88 , Amer. Math. Soc., Providence, 2008. · Zbl 1160.46001
[5]	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. · Zbl 0577.43002
[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. · Zbl 0681.43012
[7]	J. Diestel and J. J. Uhl, jr, Vector Measures , Math. Surv. 15 , Amer. Math. Soc., Providence, 1977.
[8]	E. G. Effros and Z.-J. Ruan, Operator Spaces , London Math. Soc. Monogr. N.S. 23 , Clarendon Press/Oxford Univ. Press, New York, 2000.
[9]	E. Guentner, N. Higson, and S. Weinberger, The Novikov conjecture for linear groups , Publ. Math. Inst. Hautes Études Sci. 101 (2005), 243-268. · Zbl 1073.19003
[10]	U. Haagerup, Group C \ast -algebras without the completely bounded approximation property , preprint, 1986.
[11]	U. Haagerup and J. Kraus, Approximation properties for group C \ast -algebras and group von Neumann algebras , Trans. Amer. Math. Soc. 344 (1994), 667-699. · Zbl 0806.43002
[12]	U. Haagerup, T. Steenstrup, and R. Szwarc, Schur multipliers and spherical functions on homogeneous trees , Internat. J. Math. 21 (2010), 1337-1382. · Zbl 1272.43003
[13]	M. Junge, Applications of Fubini’s theorem for noncommutative L p spaces , preprint, 2004. · Zbl 1077.46056
[14]	M. Junge, N. J. Nielsen, Z.-J. Ruan, and Q. Xu, \(\mathscr{C}\mathscr{O}\mathscr{L}_{p}\) spaces-the local structure of non-commutative L p spaces , Adv. Math. 187 (2004), 257-319. · Zbl 1084.46049
[15]	M. Junge and Z.-J. Ruan, Approximation properties for noncommutative L p -spaces associated with discrete groups , Duke Math. J. 117 (2003), 313-341. · Zbl 1042.46030
[16]	E. Kirchberg, On nonsemisplit extensions, tensor products and exactness of group C \ast -algebras , Invent. Math. 112 (1993), 449-489. · Zbl 0803.46071
[17]	E. Kissin and V. S. Shulman, Operator multipliers , Pacific J. Math. 227 (2006), 109-141. · Zbl 1130.46033
[18]	V. Lafforgue, Un renforcement de la propriété ( T ), Duke Math. J. 143 (2008), 559-602. · Zbl 1158.46049
[19]	V. Lafforgue, “Propriété ( T ) renforcée et conjecture de Baum-Connes” in Quanta of Maths , Clay Math. Proc. 11 , Amer. Math. Soc., Providence, 2010, 323-345. · Zbl 1216.19007
[20]	V. Lafforgue, Un analogue non archimédien d’un résultat de Haagerup et lien avec la propriété ( T ) renforcée , preprint, 2010, (accessed 3 August 2011).
[21]	V. Losert, On multipliers and completely bounded multipliers-The case SL(2, R ), preprint, 2010.
[22]	S. Neuwirth and E. Ricard, Transfer of Fourier multipliers into Schur multipliers and sumsets in a discrete group , to appear in Canad. J. Math., preprint, [math.FA] 1001.5332v1 · Zbl 1251.47036
[23]	N. Ozawa, About the QWEP conjecture , Internat. J. Math. 15 (2004), 501-530. · Zbl 1056.46051
[24]	G. Pisier, Regular operators between non-commutative L p -spaces , Bull. Sci. Math. 119 (1995), 95-118. · Zbl 0826.46056
[25]	G. Pisier, Non-Commutative Vector Valued L p -Spaces and Completely p-Summing Maps , Astérisque 247 , Soc. Math. France, Montrouge, 1998. · Zbl 0937.46056
[26]	G. Pisier, Similarity Problems and Completely Bounded Maps , 2nd expanded ed., Lecture Notes in Math. 1618 , Springer, Berlin, 2001. · Zbl 0843.47010
[27]	G. Pisier, Introduction to Operator Space Theory , London Math. Soc. Lecture Note Ser. 294 , Cambridge Univ. Press, Cambridge, 2003. · Zbl 1093.46001
[28]	A. Szankowski, On the uniform approximation property in Banach spaces , Israel J. Math. 49 (1984), 343-359. · Zbl 0578.46013