Journal of Mathematical Analysis and Applications 254, 645᎐653Ž. 2001 doi:10.1006rjmaa.2000.7273, available online at http:rrwww.idealibrary.com on NOTE Complemented Subspaces of Spaces of Multilinear Forms and Tensor Products1 Felix´´ Cabello Sanchez Departamento de Matematicas,´ Uni¨ersidad de Extremadura, A¨enida de El¨as, 06071 Badajoz, Spain E-mail: [email protected] Submitted by J. Hor¨ath´ Received March 13, 2000 Among other things, we show that Lq is isomorphic to a complemented = иии = G subspace of the space of multilinear forms on LppL , where q 1is r q иии q r q r s 1 n given by 1 p1 1 pn 1 q 1. The proof strongly depends on the Lϱ- ᮊ module structure of the spaces Lp. 2001 Academic Press Key Words: Banach space; multilinear form; tensor product; complemented subspace; Banach module; Banach algebra; Kothe¨ space. 1. INTRODUCTION AND SAMPLE RESULT This note stems from a misreading ofwx 2, 3 although I hope this is not entirely obvious. Our main ‘‘concrete’’ result is the following. r THEOREM 1. Let p1,..., pn and q be numbers such that 1 p1 q иии q r q r s F F ϱ ␮ 1 pniq1 q 1, with 1 p , q . Then L Ž.is a comple- mented subspace of the space of multilinear forms on L Ž.␮ = иии = L Ž.␮ , pp1 n for e¨ery ␴-finite measure ␮. That L s L Ž.⍀, ␮ is a subspace of L ŽL ,...,L .is surely well qq p1 pn known. Indeed, let f be fixed in Lq. Then we can define an n-linear form on L = иии = L by pp1 n g = иии = ¬ ␮ Žx1 ,..., xnp .L L p ftx Ž.1 Ž. t... xtdn Ž. Ž.t . 1 n H⍀ 1 Supported in part by DGICYT Project PB-97-0377. 645 0022-247Xr01 $35.00 Copyright ᮊ 2001 by Academic Press All rights of reproduction in any form reserved. 646 NOTE 555555и иии F иии According to Holder¨ inequality, one has f x1 xn 1 fxqp1 1 55 = иии = xn p , which shows that the norm of f acting as a form on LppL n 55 1 n is at most f q. As for the reverse inequality, we may and do assume that r 55s s q p i F F f is nonnegative, with f q 1. Taking xi f for 1 i n, we see 55s that xi p i 1 and иии ␮ s q ␮ s ftxŽ.1 Ž. t xtdn Ž. Ž.t f d 1, HH⍀⍀ 55s so that f 1, as a multilinear form. This shows that Lq is isometrically r isomorphic to a closed subspace of L Ž.Lpp,...,L , provided 1 p1 q иии q r q r s 1 n 1 pn 1 q 1. The proof that L is a complemented subspace of L Ž.L ,...,L will qpp1 n require an intrinsic description of the multilinear forms induced by Lq- functions in terms of certain structural properties of the spaces Lp viewed as Lϱ-modules. Then the desired projection is obtained from a standard averaging technique. 2. MULTILINEAR FORMS ON BANACH MODULES Let X1,..., Xn beŽ. Banach modules over the same Ž commutative, s ␮ Banach.Ž algebra A. The typical situation will be A Lϱ Ž.and each Xi a Kothe¨ function space on ␮. Seewx 5, 7 for information on Banach modules and Kothe¨ spaces, respectively.. We are interested in those = иии = ª ދ multilinear forms f : X1 Xn that are balancedŽ with respect to the module structure of the spaces Xi. in the sense of satisfying s fxŽ.Ž.1 ,...,axijn,..., x ,..., x fx1 ,..., xijn,...,ax ,..., x F F g g for each 1 i, j n and all a A, xkkX . The set of all these f is obviously a closed subspace of L Ž.X1,..., Xn which we denote by LAŽ.X1,..., Xn . The following result yields a useful characterization of these forms LAŽ.X1,..., Xn . = иии = LEMMA 1. Let Xi be A-modules. For a multilinear form f : X1 Xn ª ދ, the following are equi¨alent: g Ž.a f LA ŽX1,..., Xn .. y = иии = ª U Ž.b The associated Ž n 1- .linear operator X1 Xny1 Xisn an A-module homomorphism in each ¨ariable. m иии m ª U Ž.c The associated linear operator X1 ˆˆXny1 Xn is a homo- morphism ofŽ. A mˆˆиии m A -modules. NOTE 647 Proof. This is a straightforward verification which is left to the reader. U s ŽThe A-module structure of Xn is given by ²:²:ax*, x x*, ax for g g g U m иии m a A, x Xnn, x* X . The structure of a module over A ˆˆA in m иии m U m иии m и m иии m X1 ˆˆXny1 and Xn is given by Ž.Ž.a1 any11x xny1 s m иии m m иии m и U s иии и U ax11 axny1 ny11and Ž.a any1 xn Ž.a1 any1 xn , respectively.. An element z of an A-module X will be called cyclic if the set A и z s Ä4az : a g A is dense in X.Ž Clearly, if X is a minimal Kothe¨ function space on a ␴-finite measure ␮, then every nonvanishing function in X is a cyclic element for the natural LϱŽ.␮ -module structure of X. The presence of cyclic elements greatly simplifies the determination of g F F y balanced forms. Suppose ziiX ,1 i n 1, are cyclic elements over g m иии m ª U A and let f LAŽ.X1,..., Xn .IfT : X1 ˆˆXny1 Xn is the associ- m иии m ated operator, then we can recover f from the functional TzŽ 1 g UUs m иии m zny1. Xnn. Indeed, if x TzŽ.1 zny1 , then s U fazŽ.11,...,azxny1 ny1 nn² x , a1 ...axny1 n:. So, at least in the pure linear sense, LAŽ.X1,..., Xn can be seen as a U linear subspace of Xn . ␴ COROLLARY 1. Let Xi be minimal Kothe¨ function spaces on a -finite ⍀ ␮ ␮ ␴ measure space Ž., . Suppose either that is finite and Xn is -order ␴ g continuous or that each Xi is -order continuous. Then, for each f Ž. Ž. LLϱŽ ␮. X1,..., Xn , there exists a essentially unique measurable, locally integrable function g : ⍀ ª ދ such that s ␮ fxŽ1 ,..., xn .gtx Ž.1 Ž. t... xtdn Ž. Ž.t . H⍀ ¨ ␮ X Moreo er, if is finite, then g belongs to Xn. Proof. Let us recall here that a Kothe¨ function space X is ␴-order continuous if every order bounded increasing sequence converges in X. ␴ Ž.Thus, Lp is -order continuous if and only if p is finite. It is well known that X is ␴-order continuous if and only if X* s XЈ; that is, every bounded linear functional x*on X can be written as ²:x*, x s H⍀ gtxtŽ. Ž. d␮ Ž.t , for some Ž obviously locally integrable . measurable g. ␮ ␮ ; Suppose is finite. Then LϱŽ. Xi and f must be given by s U иии faŽ.1 ,...,ann² xa1 an:, 648 NOTE g ␮ U for ai LϱŽ.. On the other hand, xn is representable as an integral, so g U there is g Xn such that s иии ␮ fxŽ1 ,..., xn .gtx Ž.1 Ž. t xtdn Ž. Ž.t , H⍀ g ␮ ␮ for xi LϱϱŽ.. Since L Ž.is dense in each Xi the same representation g holds for all xiiX . As for the ␴-finite case, write ⍀ as an increasing union of measurable ⍀ ␴ subsets kiof finite measure. Since each X is -order continuous D ⍀ ⍀ ⍀ kiX Ž. kis dense in X i. Now, consider X i Ž. kas an Lϱ Ž.k -module in ⍀ ⍀ the obvious way. Taking into account that LϱŽ.k is an ideal in Lϱ Ž.we ⍀ ª ދ infer the existence of measurable functions gkk: such that s ␮ fxŽ1 ,..., xnk .gtxt Ž.1 Ž.... xtdn Ž. Ž.t , H⍀ k g ⍀ provided xiikX Ž.. Since all these g kcoincide on the common domain they define a locally integrable function g : ⍀ ª ދ in such a way that s ␮ fxŽ1 ,..., xn .gtx Ž.1 Ž. t... xtdn Ž. Ž.t H⍀ g D ⍀ g holds for every xikikX Ž.and, therefore, for every x iiX . s ␮ s ␮ ␮ ␴ COROLLARY 2. Let XipL Ž.and A Lϱ Ž., with a -finite - ϱ i measure and pnA. Then L Ž.X1,..., Xn is the space of multilinear forms ¨ r q иии q r q r s induced by Lq-functions, where q is gi en by 1 p1 1 pn 1 q 1. ␮ In particular LAŽ.X1,..., Xnq is isometrically isomorphic to L Ž.. F F Proof. If pi is finite for all 1 i n, this is a straightforward conse- quence of Corollary 1 and the computations of Section 1. The general case then follows from the obvious fact that LAŽ.A, X1,..., Xn can be identi- fied with LAŽ.X1,..., Xn if A is unital. The isomorphism is given as g = иии = follows: for f LAŽ.A, X1,..., Xn define a balanced form on X1 ¬ g Xn by Ž.Ž.x1,..., xn f 1, x1,..., xn . The inverse map transforms g ¬ LAŽ.Ž.Ž.X1,..., Xn into a, x1,..., xn gax1,..., xn . s s U Remark 1. Of course, if Xi Lϱ for all i, then LAŽ.X1,..., Xn Lϱ, which contains a complemented copy of L1. Corollary 2 remains true even if ␮ fails to be ␴-finite. This follows from a classical result of Maharam about Boolean algebras and the obvious fact that our arguments still work for strictly localizable measures.