THE MULTIPLIERS OF A p ω(g) Serap ÖZTOP

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1 THE MULTIPLIERS OF A ω() Sera ÖZTOP Préublication de l Institut Fourier n 621 (2003) htt://www-fourier.ujf-grenoble.fr/reublications.html Résumé Soit un groue localement comact abélien, dx sa mesure de Haar et ω un oids de Beurling sur. Ce travail est l étude des esaces A ω() des A ()-modules, et leurs multilicateurs. Abstract Let be a locally comact abelian grou with Haar measure d x and ω be a Beurling s weight function on. This study is concerned with the sace A ω(), A ()-modules, and their multiliers. I. Introduction and Preliminaries Let be a locally comact abelian grou with Haar measure dx and the dual grou Ĝ. A Banach sace (B, B ) is called a Banach -module, if there exists a grou reresentation L of to the grou of isometries of B with L e = I B (identity) and L x+y = L x L y. Let (A, A ) be a Banach algebra, A Banach sace (B, B ) is called a Banach A module if there exists an algebra reresentation T (continuous) of A to BL(B), the algebra of all continuous linear oerators from B to B with T a a A T a 1 a 2 = T a 1 T a 2 where is the multilication on A. For b B, T a (b) is denoted by a b. Such a module is order free if 0 is the only b B for which a b = 0 for all a A A Banach A-module B is called essential if the closed linear san of A B coincides with B. If the Banach algebra (A, A ) contains bounded aroximate identity, i.e. a bounded net 2000 Mathematics Subject Classification: 43A15, 43A20, 43A22. Keywords: Multilier, centralize, homomorhism, weight function, Banach module. 1

2 (u α ) α I such that lim u α a a A = 0 for all a A then a Banach A-module B is an essential α one, by Cohen s factorization theorem, if and only if lim u α b b B = 0 for all b B (for α more details see[4], [10]). If B is Banach A-module Hom A (B) = {T BL(B) a A, b B, T (a b) = a T (b)} is the sace of all A-module homomorhisms. The elements of Hom A (B) are traditionally called multiliers from B to B. If B is a Banach -module Hom (B) = {T BL(B) x, b B, T (L x b) = (L x T )b} is the sace of all -module homomorhisms. The elements of H (B) will be called centralizer. In the most cases, a multilier is equal to a centralizer, however in the certain saces, a centralizer need not be a multilier [3, 16]. In this aer, we introduce the weighted Banach A ()-modules and discuss some multilier roblems where A () is the -Fourier algebra. The sace A (), 1 <, was introduced by Figa-Talamanca [7] and roved that the multilier sace of L () (as a centralizer) is isometrically isomorhic to the dual sace A () of A () rovided is abelian. For nonabelian locally comact grou, Eymard [6] studied the Fourier algebra A 2 () = A() and for general, 1 <, Herz [9] roved that A () is a Banach algebra under ointwise multilication. We are interested in the structure theory of the weighted A (), denoted by Aω(). Using by the technical develoed by Sector [15] we show that Aω() is a Banach A ()-module under ointwise multilication. Note that A () is a Fourier algebra under ointwise multilication but it is not an algebra under convolution. Moreover, it is obtained some necessary and sufficients conditions on oerator to be an Aω()-multiliers. II. The A ω() saces and their convolutors Let be a locally comact abelian grou with Haar measure dx and ω be a non negative continuous function on. L ω() = { f f w L ()} denote the Banach sace under the natural norm f,ω = f ω, 1 and for 1 < its dual sace is L ω 1 () with = 1. C 0,ω () denote the Banach subsace of L ω () such that f ω C 0(), the sace of all continuous functions vanishing at infinity of. If ω is a weight function, i.e. continuous function satisfying ω(x) 1, ω(x + y) ω(x)ω(y) then L 1 ω () is a Banach algebra under convolution called a Beurling algebra. It follows that Lω() is an essential Banach L 1 ω ()-module with resect to convolution [2]. x. Throughout it will be assumed that ω is even function, i.e. satisfy ω(x) = ω( x) for all 2

3 As the classical thecnic of harmonic analysis it follows that PROPOSITION II.1. Let f Lω() and g L ω 1 () where C 0,ω 1() with f g,ω 1 f,ω g,ω 1. = 1. Then f g Thus by Proosition II.1 it can be defined a bilinear ma from L ω() L ω 1 () into C 0,ω 1() by b( f, g) = f g, f (x) = f ( x) and b 1. Then b lifts a linear ma B from L ω() and L ω 1 () as a Banach sace, into C 0,ω 1() and B 1, [1, 5, 8]. DEFINITION II.2. It will be denoted by Aω() the subsace of C 0,ω 1() of all functions h satisfying h = f i g i, f i Lω(), g i L ω 1 () such that f i,ω g i,ω 1 < equied with the norm { h = inf A f ω i,ω g i,ω 1, h = f i g i }. A ω() is a Banach sace under this norm and identifies with the sace of the quotient Banach sace of L ω() γ L ω 1 () with K, i.e. where K is the kernel of the bilinear form B. A ω() L ω() γ L ω 1 ()/K Since L ω() is the reflexive Banach sace, by the general oerator theory [5], the algebra of the oerators BL(L ω(), L ω()) identifies with the dual sace (L ω() γ L ω 1 ()) and the subsace K is the annihilator of the translations it follows that Hom ω(), L ω 1 () ) ω() γ L ( ω 1 ()/K ) ( A ω() ). Because Lω() is an essential, order free L 1 ω ()-module then K is also the annihilator of the oerators acted by L 1 ω (), see for examle [3, 10, 14], so (A ω()) Hom ω(), L ω() ) Hom L 1 ω ω(), L ω() ). Since C c () = L ω(), C c () = L ω 1 () and the convolution of two suort comact functions is suort comact it follows that C c () A ω() is dense in A ω(). Also L ω() is the essential Banach L 1 ω ()-module under convolution, A ω() is an essential Banach L 1 ω ()- module [13]. So, for every u A ω() and x, x L x f (y) = f x (y) = f (y x) for all y. f x is continuous function where THEOREM II.3. Let T BL(A ω(), A ω()), f L ω(), g L ω 1 () and h L 1 ω (). The following statements are equivalent 3

4 (i) T (h f g) = h T ( f g) (ii) T ((L x f ) g) = L x T ( f g) (iii) T BL(Lω(), Lω()) such that T ( f g) = ( T f ) g. Proof. (i) (ii): since Aω() is the essential Banach L 1 ω ()-module, it is obtained the equivalence of (i) and (ii). (iii) (i): it is obvious. (ii) (iii): let T be in BL(A ω(), A ω()), f L ω() the ma g L ω 1 () (T ( f g))(e) defines a linear, continuous functional. Because of the reflexivity of L ω() there exists an element T f of L ω() such that and it is obtained the following equivality T f, g = ( T ( f g) ) (e) ( T ( f g) ) (x) = T f, L x. COROLLARY II.4. Hom L 1 ω ( A ω(), A ω() ) = Hom L 1 ω ω(), L ω() ). Proof. It is an interretation of the equivalence of (ii) and (iii). III. The multiliers of A ω() If ω = 1, Herz [9] roved that A () is a Banach algebra under ointwise multilication and A () has a bounded aroximate identity, i.e. there exists (α i ) i I A () such that lim i u α i u A = 0 for all u A (). Using by the roof of Herz s theorem given by Sector [15] the next result follows. THEOREM III.1. A ω() is an essential, order free Banach A ()-module under ointwise multilication. Then Proof. Let us consider two functions u = f g and v = h k where f, g, j, k C c (). u A ω v A ω f,ω g,ω 1 h,ω k,ω 1 u(x) = f (y 1 x 1 )g(y 1 ) dy and v(x) = h(z 1 x 1 )k(z 1 ) dz 4

5 we have (uv)(x) = f (y 1 x 1 )g(y 1 ) dy where A z = ã z b z, a z, b z C c () h(z 1 y 1 x 1 )k(z 1 y 1 ) dz = A z (x) dz a z (x) = f (x)h(z 1 x) b z (x) = g(x)k(z 1 x) since the maing z A z is continuous as the Banach valued integral, it sufficies to show that But A z A ω dz f,ω g,ω 1 h k. A z dz A a ω z,ω b z,ω 1 dz ( a z ) 1/ (,ω dz = ( f (x) ω(x) dx ( g(x) ω 1 (x) dx = f,ω g,ω 1 h k. So, it is obtained that for u A () and v A ω() uv A ω u A v A ω. ) b z 1/,ω 1 dz h(z 1 x) ) 1/ dz ) 1/ k(t ) dt Since C c () A ω() is dense in A ω() and A ω() is a Banach A ()-module it is obtained that A ω() is an essential Banach module and order free, i.e. u A (), uv = 0 for all v A ω() then v = 0. Remark. Since A 2 () A r () A () for 1 < < r < 2 or 2 is also A r ()-module for 1 < < r 2 or 2 r < <, i.e. r < <, A ω() A r () A ω() A ω(). In articular, for r = 2, since A 2 () identifies withl 1 (Ĝ) = F (L1 (Ĝ)), A ω() has a structure of L 1 (Ĝ)-module and the action ϕ is given by ϕ h A ω() for ϕ L 1 (Ĝ), h A ω(). A (). PROPOSITION III.2. A ω() has a comatible structure Ĝ-module with the action of Proof. Since Aω() is the essential Banach L 1 (Ĝ)-module A ω() has a unique comatible structure Ĝ-module [11]. This action is given by γ Ĝ, x, u A ω(), (M γ u)(x) = γ(x) u(x). Indeed, γ 1, γ 2 Ĝ 5

6 (M γ1 +γ 2 u)(x) = (γ 1 + γ 2 )(x) u(x) = γ 1 (x)γ 2 (x) u(x) = (M γ1 M γ2 u)(x) M e u = 1 u = u M γ u u M γ M γ u M γ u. So M γ is an isometry and also L γ ϕ u = M γ ϕ u = ϕ(m γ u) for ϕ L 1 (Ĝ). By the density of A 2 () in A () it follows (M γ f )u = M γ ( f u) = f M γ (u), f L 1 (Ĝ). THEOREM III.3. Let be a locally comact abelian grou and Ĝ be a dual grou, 1 < < for a continuous linear oerator T the following statements are equivalent (i) For all f L 1 (Ĝ) and u A ω() such that f u = 0 then f T (u) = 0. (ii) T is L 1 (Ĝ)-module homomorhism (or multilier) i.e. T Hom L 1 (Ĝ)(A ω(), A ω()). Proof. Since Aω() 1 is an essential and order free Banach L (Ĝ)-module, it is obtained from [12]. PROPOSITION III.4. For 1 < r 2 or 2 r <, Hom L 1 (Ĝ) ( A ω(), Aω() ) ( = HomĜ A ω(), Aω() ) ( = Hom Ar A ω(), Aω() ). So we consider that each multilier on A ω() as a centralizer. Acknowledgment. I would like to thank Prof. Dr.. Muraz for helful suggestions regarding this aer. References [1] BONSALL F.F., DUNCAN J., Comlete normed algebras, Sringer, Berlin, [2] BRAUN W., FEICHTINER H.., Banach saces of distributions having two module structures, J. Func. Anal., 51 (1983), [3] DATRY C., MURAZ., Analyse harmonique dans les modules de Banach, I : roriétés générales, Bull. Sci. Math., 119 (1995), [4] DORAN R.S., WICHMANN J., Aroximate identities and factorization in Banach modules, Lecture Notes Mathematics, 768, Sringer, Berlin,

7 [5] DUNFORD N., SCHWARTZ J.T., Linear Oerators, I, Interscience Publisher, New York, [6] EYMARD P., L algèbre de Fourier d un groue localement comact, Bull. Soc. Math., 92 (1964), [7] FIER-TALAMANCER A., Multiliers of -integrable functions, Bull. Amer. Math. Soc., 70 (1964), [8] ROTHENDIECK A., Produits tensoriels toologiques et esaces nucléaires, American Mathematical Society, Providence, Rhode Island (1955). [9] HERZ C., The theory of -saces with an alication to convolution oerators, Trans. Amer. Math. Soc., 154 (1971), [10] HEWIT E., ROSS K.A., Abstract Harmonic Analysis, I, Sringer, Berlin, [11] LIU T.S., VAN ROOIJ A. & WAN J.K., rou reresentations in Banach saces: orbits and almost eriodicity, Studies and essays resented to Yu-Why-Chen, Taiei, Math. Research center (1970), [12] MURAZ., Multilicateurs sur les L 1 ()-modules, roue de travail d Analyse harmonique, Institut Fourier, renoble, [13] ÖZTOP S., ÜRKANLI A.-T., Multiliers and tensor roducts of weighted L -saces, Acta Math. Sci., 21-B(1) (2001), [14] RIEFFEL M.A., Multiliers and tensors roducts of L -saces of locally comact grous, Studia Math., 33 (1969), [15] SPECTOR R., Sur la structure locale des groues abéliens localement comacts, Bull. Soc. Math., 24 (1970), [16] TEWARI U.B., DUTTA & VAIDYA J.P., Multiliers of grou algebra of vector-valued functions, Proc. Amer. Math. Soc., 81 (1981), Sera ÖZTOP Istanbul University Faculty of Sciences Deartment of Mathematics, Vezneciler, ISTANBUL (Turkey) & INSTITUT FOURIER Laboratoire de Mathématiques UMR5582 (UJF-CNRS) BP St MARTIN D HÈRES Cedex (France) seraoz@mozart.ujf-grenoble.fr seraozto@hotmail.com 7