Normes de Sobolev et convoluteurs bornés sur \(L^ 2(G)\). (Sobolev norms and bounded convolvers on \(L^ 2(G)\)). (French) Zbl 0734.43002

Jolissaint, Paul; Valette, Alain

Ann. Inst. Fourier 41, No. 4, 797-822 (1991).

A locally compact group G equipped with a length-function L has property (RD) with respect to L if any rapidly decreasing function on G defines a bounded convolver on \(L^ 2(G)\). We give a fairly general sufficient condition for the pair (G,L) to have property (RD). For such a pair, we characterize positive definite functions on G that are weakly associated to the left regular representation and, in the discrete case, we deal with approximation properties of the Fourier algebra of G.

Reviewer: A.Valette (Neuchâtel)

MSC:

43A35	Positive definite functions on groups, semigroups, etc.
43A15	\(L^p\)-spaces and other function spaces on groups, semigroups, etc.
43A70	Analysis on specific locally compact and other abelian groups
46L05	General theory of \(C^*\)-algebras

Keywords:

locally compact group; length-function; rapidly decreasing function; bounded convolver; positive definite functions; left regular representation; Fourier algebra

References:

[1]	[A] , Subalgebras of C*-algebras, Acta Math., 123 (1969), 141-224. · Zbl 0194.15701
[2]	[Bor] , Compact Clifford-Klein forms of symmetric spaces, Topology, 2 (1963), 111-122. · Zbl 0116.38603
[3]	[Bou] , Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Hermann, 1968.
[4]	[Boz] , Remark on Herz-Schur multipliers on free groups, Math. Ann., 258 (1981), 11-15. · Zbl 0483.43004
[5]	[BozJaSp] , et , Infinite Coxeter groups do not have Kazhdan’s property (T), J. Operator Theory, 19 (1988), 63-67. · Zbl 0662.20040
[6]	[dCaHaa] et , Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. J. Math., 107 (1985), 455-500. · Zbl 0577.43002
[7]	[CoMos] et , Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology, 29 (1990), 345-388. · Zbl 0759.58047
[8]	[CoHaa] et , 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
[9]	[CowHaaHo] , et , Almost L2 matrix coefficients, J. reine angew. Math., 387 (1988), 97-110. · Zbl 0638.22004
[10]	[E] , L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France, 92 (1964), 181-236. · Zbl 0169.46403
[11]	[FHarz] et , Distances hilbertiennes invariantes sur un espace homogène, Ann. Inst. Fourier, 24-3 (1974), 171-217. · Zbl 0265.43013
[12]	[GhHarp] Sur les groupes hyperboliques d’après Mikhael Gromov, ouvrage collectif édité par E. Ghys et P. de la Harpe, Progress in Math. 83, Birkhäuser 1990. · Zbl 0731.20025
[13]	[G] , Hyperbolic groups, in “Essays in group theory”, édité par S. M. Gersten (Springer, 1987), 75-263.
[14]	[Haa] , An example of a non-nuclear C*-algebra which has the metric approximation property, Invent. Math., 50 (1979), 279-293. · Zbl 0408.46046
[15]	[dHarp1] , Groupes de Coxeter infinis non affines, Expo. Math., 5 (1987), 91-96. · Zbl 0605.20049
[16]	[dHarp2] , Groupes hyperboliques, algèbres d’opérateurs et un théorème de Jolissaint, C.R. Acad. Sci. Paris, Sér. I, 307 (1988), 771-774. · Zbl 0653.46059
[17]	[dHarpV] et , La propriété (T) de Kazhdan pour les groupes localement compacts, Astérisque 175, Soc. Math. France, 1989. · Zbl 0759.22001
[18]	[Hel] , Groups and geometric analysis, Academic Press, 1984.
[19]	[Her] , Une généralisation de la notion de transformée de Fourier-Stieltjes, Ann. Inst. Fourier, Grenoble, 24 (1974), 145-157. · Zbl 0287.43006
[20]	[Jo1] , Rapidly decreasing functions in reduced C*-algebras of groups, Trans. Amer. Math. Soc., 317 (1990), 167-196. · Zbl 0711.46054
[21]	[Jo2] , K-theory of reduced C*-algebras and rapidly decreasing functions on groups, K-theory, 2 (1989), 723-735. · Zbl 0692.46062
[22]	[L] , Tensor products and nuclear C*-algebras, in “Operator algebras and applications”, Proc. Symp. Pure Math., 38 (vol. 1), Amer. Math. Soc., 1982, 379-399. · Zbl 0498.46044
[23]	[Ma] , On the faithful representations of infinite groups by matrices, Amer. Math. Soc. Transl., 45 (1965), 1-18. · Zbl 0158.02905
[24]	[Mou] , Hyperbolic Coxeter groups, PhD thesis, Ohio State University, 1988.
[25]	[Re] , Centralizers of the Fourier algebra of an amenable group, Proc. Amer. Math. Soc., 32 (1972), 539-542. · Zbl 0258.43010
[26]	[Ro] , A non-extendible positive map on the reduced C*-algebra of a free group, Bull. London Math. Soc., 18 (1986), 389-391. · Zbl 0614.46048
[27]	[RoSm] et , Liftings and extensions of maps on C*-algebras, J. Operator Theory, 21 (1988), 117-131. · Zbl 0699.46041
[28]	[ScWal] et , Topological methods in group theory, in «Homological group theory», édité par C.T.C. Wall, London Math. Soc. Lecture Notes Series 36, Cambridge Univ. Press, 1979. · Zbl 0423.20023
[29]	[Sk] , Approche de la conjecture de Novikov par la cohomologie cyclique [d’après A. Connes, M. Gromov et H. Moscovici], Sém. Bourbaki, Fév. 1991, Exposé 739. · Zbl 0749.58056
[30]	[St] , Extension of positive maps into B(H), J. Funct. Anal., 66 (1986), 235-254. · Zbl 0637.46061
[31]	[17] , Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen, Inst. Hautes Etudes Sci. Publ. Math., 5 (1960). · Zbl 0158.32901
[32]	[T2] , Reductive groups over local fields, Proc. Symp. Pure Math. 33 (vol. 1), Amer. Math. Soc., 1979, 29-69. · Zbl 0415.20035
[33]	[V] , Weak forms of amenability for split rank 1 p-adic groups, pré-publication 1990, à paraître dans «p-adic methods and applications», Oxford University Press.
[34]	[Was] , On tensor products of certain group C*-algebras, J. Funct. Anal., 23 (1976), 239-254. · Zbl 0358.46040