A NOTE ON MULTIPLIERS OF L p (G, A)

Transcription

1 J. Aust. Math. Soc. 74 (2003), A NOTE ON MULTIPLIERS OF L p (G, A) SERAP ÖZTOP (Received 8 December 2000; revised 25 October 200) Communicated by A. H. Dooley Abstract Let G be a locally compact abelian group, < p <, and A be a commutative Banach algebra. In this paper, we study the space of multipliers on L p.g; A/ and characterize it as the space of multipliers of certain Banach algebra. We also study the multipliers space on L.G; A/ L p.g; A/ Mathematics subject classification: primary 43A22.. Introduction and preliminaries Let G be a locally compact abelian group with Haar measure, A be a commutative Banach algebra with identity of norm. Denote by L.G; A/ the space of all Bochner integrable A-valued functions defined on G. It is a commutative Banach algebra under convolution and has an approximate identity in C c.g; A/ of norm, L p.g; A/ is the set of all strong measurable functions f : G A such that f.x/ p A is integrable for p <, thatis, f.x/ p A L.G/. The norm of a function f in L p.g; A/ is defined as ( ) =p f L p.g;a/ = f.x/ p A dx p < : G It follows that L p.g; A/ is a Banach space for p < and L p.g; A/ is an essential L.G; A/-module under convolution such that for f L.G; A/ and g L p.g; A/, we have f g L p.g;a/ f L.G;A/ g L p.g;a/ : This work was supported by the Research Fund of Istanbul University. Project No. B 966/ c 2003 Australian Mathematical Society /03 $A2:00 + 0:00 25

2 26 Serap Öztop [2] Denote by C c.g; A/ the space of all A-valued continuous functions with compact support. C c.g; A/ is dense in L p.g; A/ (for more details see [2, 6, 7]). For each f L.G; A/, define the mapping T f by T f.g/ = f g whenever g L p.g; A/. T f is an element of `.L p.g; A//, Banach algebra of all continuous linear operators from L p.g; A/ to L p.g; A/, and T f f L.G;A/. Identifying f with T f, we get an embedding of L.G; A/ in `.L p.g; A//. LetH L.G;A/.L p.g; A// denote the space of all module homomorphisms of L.G; A/-module L p.g; A/, that is, an operator T `.L p.g; A// satisfies T. f g/ = f T.g/ for each f L.G; A/, g L p.g; A/. The module homomorphisms space, called the multipliers space H L.G;A/.L p.g; A//; is an essential L.G; A/-module by. f T /.g/ = f T.g/ = T. f g/ for all g L p.g; A/. Let A be a Banach algebra without order, for all x A, xa = Ax ={0} implies x = 0. Obviously if A has an identity or an approximate identity then it is without order. A multiplier of A is a mapping T : A A such that T. fg/ = ft.g/ =.Tf/g;.f; g A/: By M.A/ we denote the collection of all multipliers of A. Every multiplier turns out to be a bounded linear operator on A. IfA is a commutative Banach algebra without order, M.A/ is a commutative operator algebra and M.A/ is called the multiplier algebra of A [5]. In this paper we are interested in the relationship between the multipliers L.G; A/- module and the multipliers on a certain normed (or Banach) algebra. The multipliers of type.p; p/ and multipliers of the group L p -algebras were studied and developed by many authors. Let us mention McKennon [0, ] Griffin[5], Feichtinger [3] and Fisher [4]. In these studies, a multiplier is defined to be an invariant operator (a bounded linear operator T commutes with translation). In the case of a scalar function space on G, the multipliers are identified with the translation invariant operators. However, in the Banach-valued function spaces, an invariant operator does not need to be a multiplier [8, 4]. Dutry [] gave a new proof of the identification theorem concerning multipliers of L.G/-module and of Banach algebra. His ideas are used in this paper for the generalization of the results of McKennon concerning multipliers of type.p; p/ to the Banach-valued function spaces. We briefly describe the content of this paper. In Section 2 we construct the p- temperate functions space for the Banach-valued function spaces whenever < p < and study their basic properties. In Section 3 we characterize the multipliers space of L p.g; A/ as a certain Banach algebra and extend the results of McKennon

3 [3] A note on multipliers of L p.g; A/ 27 to Banach-valued space. In Section 4 we study the multipliers space of L.G; A/ L p.g; A/. 2. The L t p (G, A) space and its basic properties Let G be a locally compact abelian group with Haar measure, A a commutative Banach algebra with identity of norm. or DEFINITION 2.. An element f L p.g; A/ is called p-temperate function if f t L p.g;a/ = sup{ g f L p.g;a/ g L p.g; A/; g L p.g;a/ } < f t L p.g;a/ = sup{ g f L p.g;a/ g C c.g; A/; g L p.g;a/ } < : The space of all such f is denoted by L t p.g; A/. It is easy to see that ( ) L t.g; A/; p t L p.g;a/ is a normed space. For each f L t p.g; A/, there is precisely one bounded linear operator on L p.g; A/, denoted by W f, such that (2.) W f.g/ = g f and W f = f t L p.g;a/ : It is easy to check that W f H L.G;A/.L p.g; A//. PROPOSITION 2.2. L t p.g; A/ is a dense subspace of L p.g; A/. PROOF. Since each f C c.g; A/ belongs to L t p.g; A/ and C c.g; A/ is dense in L p.g; A/, the proof is completed. LEMMA 2.3. The space L t p.g; A/ is a normed algebra under the convolution. PROOF. By (2.) we get f g t = sup h. f g/ L p.g;a/ L p.g;a/ = sup W g.h f / L p.g;a/ h L p.g;a/ h L p.g;a/ W g sup h f L p.g;a/ = g t f L h p t.g;a/ L p.g;a/ for all f and g in L t.g; A/. Hence p.lt.g; A/; p t / is a normed algebra. L p.g;a/ Let us notice that (2.2) W f g = W f W g = W g W f for all f and g in L t p.g; A/. Moreover, the closed linear subspace of `.L p.g; A// spanned by {W f g f L t p.g; A/; g C c.g; A/} is denoted by 3 L p.g;a/.

4 28 Serap Öztop [4] THEOREM 2.4. The space 3 L p.g;a/ is a complete subalgebra of H L.G;A/.L p.g; A// and it has a minimal approximate identity, that is, a net.t / such that lim T and lim T T T =0 for all T 3 L p.g;a/. PROOF. Let f L t p.g; A/, then W f `.L p.g; A//. Since L p.g; A/ is a L.G; A/-module we have W f.g h/ = g h f = g W f.h/ for all g L.G; A/ and h L p.g; A/. Thus W f belongs to H L.G;A/.L p.g; A//. Since H L.G;A/.L p.g; A// is a Banach algebra under the usual operator norm, 3 L p.g;a/ is a complete subalgebra of H L.G;A/.L p.g; A//. Now we only need to prove the existence of minimal approximate identity of 3 L.G;A/. Let.8 p U / be a minimal approximate identity for L.G; A/ [2]. If.8 / denotes the product net of.8 U / with itself, then.8 / is again minimal approximate identity for L.G; A/. It is easy to see that the net W 8 is in 3 L p.g;a/ and lim W 8. Let f L t.g; A/ and g C p c.g; A/. Since.2:2/ and.8 / is a minimal approximate identity for L.G; A/, weget lim W 8 W f g W f g =lim.w 8 W g W g / W f lim W g 8 g W f lim g 8 g L.G;A/ W f =0: Consequently, we have lim W 8 T T =0for all T 3 L.G;A/. p PROPOSITION 2.5. The space 3 L p.g;a/ is an essential L.G; A/-module. PROOF. Let g L.G; A/, W f 3 L p.g;a/. Defineg W f : L p.g; A/ L p.g; A/ by letting.g W f /.h/ = W f.h g/ = W f.g h/ for each h L p.g; A/. g W f = sup W f.g h/ L p.g;a/ f t L.G;A/ g p L.G;A/: h L p.g;a/ Consequently, 3 L p.g;a/ is a L.G; A/-module. On the other hand, since L.G; A/ has a minimal approximate identity.8 /,.8 0/ with a compact support such that it is also an approximate identity in L p.g; A/, [2]. For any W f 3 L p.g;a/,wehave 8 W f W f = sup.8 W f W f /.h/ L p.g;a/ h L p.g;a/ = sup W f.8 h h/ L p.g;a/ g L p.g;a/ f t L p.g;a/ 8 h h L p.g;a/ = 0

5 [5] A note on multipliers of L p.g; A/ 29 for all h L p.g; A/. Using [3, Proposition 3.4] we have that 3 L p.g;a/ is an essential L.G; A/-module. Moreover, 3 L p.g;a/ contains L.G; A/. 3. Identification for the multipliers spaces of L (G, A)-module with the multipliers space of certain normed algebra In this section, we obtain the generalization of the results of McKennon [0,, 2] to the Banach-valued spaces. PROPOSITION 3.. Let T be in H L.G;A/.L p.g; A// and f; g L p.g; A/. Then,.i/ if f L t.g; A/, T. f / p Lt p.g; A/;.ii/ if g L t p.g; A/, T. f g/ = f T.g/. PROOF. (i) Let f be in L t p.g; A/. By the definition T H L.G;A/.L p.g; A//, T. f / t = sup{ h T. f / L p.g;a/ L p.g;a/ h C c.g; A/; h L p.g;a/ } = sup{ T.h f / L p.g;a/ h C c.g; A/; h L p.g;a/ } T f t < : L p.g;a/ HencewegetT. f / L t p.g; A/. To prove (ii), let g be in L t.g; A/. SinceC p c.g; A/ is dense in L p.g; A/, for each f L p.g; A/ there exists. f n / C c.g; A/ such that lim n f n f L p.g;a/ = 0. From (2.) we get lim n f n g f g L p.g;a/ = 0. By (i) we have lim n f n T.g/ f T.g/ L p.g;a/ = 0 and f T.g/ = lim n f n T.g/ = lim n T. f n g/ = T. f g/. DEFINITION 3.2. For the space 3 L p.g;a/, the space.3 L p.g;a// is defined by.3 L p.g;a// ={T H L.G;A/.L p.g; A// T W 3 L p.g;a/, for all W 3 L p.g;a/}: LEMMA 3.3. The space.3 L p.g;a// is equal to the space H L.G;A/.L p.g; A//. PROOF. Let T H L.G;A/.L p.g; A//. For any S 3 L p.g;a/, wehaves = W f g, for each f L t p.g; A/, g C c.g; A/. By Proposition 3. we get.t W f g /.h/ = T.h f g/ = h T. f g/ = W T. f g/.h/ = W g T. f /.h/ for all h L p.g; A/. Thus T S 3 L p.g;a/. Consequently,.3 L p.g;a// = H L.G;A/.L p.g; A//:

6 30 Serap Öztop [6] Let us note that we have the inclusion M.3 L p.g;a// H L.G;A/.3 L p.g;a//. THEOREM 3.4. Let G be a locally compact abelian group, < p <, and A be a commutative Banach algebra with identity of norm. The space of multipliers on Banach algebra 3 L p.g;a/, M.3 L p.g;a//, is isometrically isomorphic to the space.3 L p.g;a//. PROOF. Define the mapping z : 3 L p.g;a/ M.3 L p.g;a// by letting z.t / = ² T for each T 3 L p.g;a/,where² T.S/ = T S for all S 3 L p.g;a/. Note that z is well defined and moreover if ² T.S K / = T S K = ² T.S/ K for all S, K 3 L p.g;a/, ² T M.3 L p.g;a//. It is obvious that the mapping T ² T is linear. We now show that it is an isometry. We obtain easily T ² T.SinceW 8 is a minimal approximate identity for the space 3 L.G;A/, wehave p ² T = T S T W 8 sup sup T : S 3 L p.g;a/ S W 8 Therefore, ² T = T. Finally, we show that the mapping T ² T is onto. It is sufficient to show that if ² is an element of M.3 L p.g;a//, the limit of ²8 exists for the strong operator topology and this limit T satisfies ² T = ². Let² be in M.3 L p.g;a// and.8 / L.G; A/. By ²8. f g/ = ².8 f /g,wehave (3.) lim.²8 /. f g/ = ² f.g/ for all f L.G; A/, g L p.g; A/. Since L p.g; A/ is an essential L.G; A/- module, the limit of.²8 /. f g/ exists in L p.g; A/ and is denoted by Tg,and Tg H L.G;A/.L p.g; A//. From(3.) we get, for all f L.G; A/, (3.2) f T = ² f: So for all W 3 L p.g;a/ we have (3.3) T 8 W =.²8 / W = ².8 W /: Since 3 L p.g;a/ is an essential L.G; A/-module, we have T W = ².W / and also ² T.W / = ².W / for all W 3 L p.g;a/.so² T = ². COROLLARY 3.5. The following spaces of multipliers are isometrically isomorphic: M.3 L p.g;a// = H L.G;A/.L p.g; A//.

7 [7] A note on multipliers of L p.g; A/ 3 REMARK 3.6..i/ Let p =. Since L t.g; A/ is a Banach algebra, it follows that L t.g; A/ = L.G; A/ and 3 L p.g;a/ is isomorphic to L.G; A/ as a Banach algebra. Thus by [4] wegeth L.G;A/.L p.g; A// = M.L.G; A// = M.G; A/. Here M.G; A/ denotes A-valued bounded measure space..ii/ If A = c we have the case of the scalar valued function space in [0, ]. 4. The identification for the space L (G, A) L p (G, A) Before starting the identification, let us mention some properties of the space L.G; A/ L p.g; A/. If < p <, then the space L.G; A/ L p.g; A/ is a Banach space with the norm f = f L.G;A/ + f L p.g;a/ for f L.G; A/ L p.g; A/. LEMMA 4.. For L.G; A/ L p.g; A/,.i/ L.G; A/ L p.g; A/ is dense in L.G; A/ with respect to the norm L.G;A/..ii/ For every f L.G; A/ L p.g; A/ and x G, x L x f is continuous, where L x f.y/ = f.x y/ for all y G. PROOF. (i) Since C c.g; A/ is dense in L.G; A/ with respect to the norm L.G;A/ and C c.g; A/ L.G; A/ L p.g; A/ L.G; A/ it is obtained. (ii) Let f L.G; A/ L p.g; A/. It is easy to see that L x f = f.by[2] the function x L x f is continuous, G L p.g; A/, where p <. Therefore for any x G and ž>0, there exists U #.x / and U 2 #.x / such that for every x U L x f L x f L p.g;a/ <ž=2 and for every x U 2 L x f L x f L.G;A/ <ž=2: Set V = U U 2, then for all x V,wehave L x f L x f <ž. PROPOSITION 4.2. The space L.G; A/ L p.g; A/ has a minimal approximate identitiy in L.G; A/. LEMMA 4.3. The space L.G; A/ L p.g; A/ is an essential L.G; A/-module. PROOF. Let f L.G; A/ and g L.G; A/ L p.g; A/. SinceL p.g; A/ is an L.G; A/-module, we have f g = f g L.G;A/ + f g L p.g;a/ f g :

8 32 Serap Öztop [8] By [3, Proposition 3.4] we get that L.G; A/ L p.g; A/ is an essential L.G; A/- module. PROPOSITION 4.4. L.G; A/ L p.g; A/ is a Banach ideal in L.G; A/. PROPOSITION 4.5. L.G; A/ L p.g; A/ is a Banach algebra with the norm. PROOF. For any f; g L.G; A/ L p.g; A/, using the inequality we get that f g f g. L.G;A/ ; COROLLARY 4.6. The space L.G; A/ L p.g; A/ is a Segal algebra. PROOF. By Lemma 4. and Proposition 4.5 we obtain that L.G; A/ L p.g; A/ is a Segal algebra. We now return to Section 3 to mention the multipliers of L.G; A/ L p.g; A/. Since L.G; A/ L p.g; A/ is an L.G; A/-module and a Banach algebra, then we get easily M.L.G; A/ L p.g; A// = HL.G;A/.L.G; A/ L p.g; A//. PROPOSITION 4.7. H L.G;A/.L.G; A/ L p.g; A// is an essential Banach module over L.G; A/. PROOF. Let f L.G; A/ and T H L.G;A/.L.G; A/ L p.g; A//. Define the operator ft on L.G; A/ L p.g; A/ by. ft/.g/ = T. f g/ for all f L.G; A/ L p.g; A/. By Proposition 4.5 T is well defined. Then H L.G;A/.L.G; A/ L p.g; A// is an L.G; A/-module. Let.8 / be a minimal approximate identity for L.G; A/ and T be in H L.G;A/.L.G; A/ L p.g; A//. Wehave lim 8 T T =0: By [3, Proposition 3.4], we have that H L.G;A/.L.G; A/ L p.g; A// is an essential Banach module over L.G; A/. Define } to be the closure of L.G; A/ in H L.G;A/.L.G; A/ L p.g; A// for the operator norm. Evidently, H L.G;A/.L.G; A/ L p.g; A// =.H L.G;A/.L.G; A/ L p.g; A/// e = } =.}/ e ; where. / e denotes the essential part and we have H L.G;A/.L.G; A/ L p.g; A// =.}/:

9 [9] A note on multipliers of L p.g; A/ 33 Here.}/ is defined as the space of the elements T H L.G;A/.L.G; A/ L p.g; A// such that T } }. Using the same method as in Theorem 3.4 we get the following lemma. LEMMA 4.8. The multipliers space of Banach algebra } is isometrically isomorphic to the space.}/. We also get the following corollary. COROLLARY 4.9. H L.G;A/.L.G; A/ L p.g; A// = M.}/. So the multipliers space of L.G; A/ L p.g; A/ can be identified with the multipliers space of the closure of L.G; A/ in H L.G;A/.L.G; A/ L p.g; A//. REMARK 4.0..i/ It is evident that every measure ¼ M.G; A/ defines multiplier for L.G; A/ L p.g; A/, < p <. This is obvious from the fact that ¼ f ¼ f, f L.G; A/ L p.g; A/. On the other hand, for ¼ M.G; A/,wehave¼ L.G; A/ L.G; A/, the inclusion in the space H L.G;A/.L.G; A/ L p.g; A//. Hence, ¼ } }, thus M.G; A/ can be embeded into.}/..ii/ If A = candg is a noncompact locally compact abelian, we have the more general result than the Corollary 3.5. in Larsen [9]. References [] C. Datry, Multiplicateurs d un L.G/-module de Banach consideres comme multiplicateurs d une certaine algèbre de Banach (Groupe de travail d Analyse Harmonique, Institut Fourier, Grenoble, 98). [2] N. Dinculeanu, Integration on locally compact spaces (Nordhoff, The Netherlands, 974). [3] H. Feichtinger, Multipliers of Banach spaces of functions on groups, Math. Z. 52 (976), [4] M. J. Fisher, Properties of three algebras related to L p -multipliers, Bull. Amer. Math. Soc. 80 (974), [5] J. Griffin and K. McKennon, Multipliers and group L p algebras, Pacific J. Math. 49 (973), [6] G. P. Johnson, Spaces of functions with values in a Banach algebra, Trans. Amer. Math. Soc. 92 (959), [7] H. C. Lai, Multipliers of Banach-valued function spaces, J. Austral. Math. Soc. (Ser. A) 39 (985), [8] H. C. Lai and T. K. Chang, Multipliers and translation invariant operators, Tohoku Math. J. 4 (989), 3 4. [9] R. Larsen, An introduction to the theory of multipliers (Springer, Berlin, 97). [0] K. McKennon, Multipliers of type.p; p/, Pacific J. Math. 43 (972),

10 34 Serap Öztop [0] [], Multipliers of type.p; p/ and multipliers of group L p algebras, Pacific J. Math. 45 (972), [2], Corrections to [0, ] and [5], Pacific J. Math. 6 (975), [3] M. Rieffel, Induced Banach representations of Banach algebras and locally compact groups, J. Funct. Anal. (967), [4] U. B. Tewari, M. Dutta and D. P. Vaidya, Multipliers of group algebra of vector-valued functions, Proc. Amer. Math. Soc. 8 (98), [5] J. K. Wang, Multipliers of commutative Banach algebras, Pacific J. Math. (96), Istanbul University Faculty of Sciences Department of Mathematics Vezneciler/Istanbul Turkey serapoztop@hotmail.com, oztops@istanbul.edu.tr