NOTE Complemented Subspaces of Spaces of Multilinear Forms And

Home , Complemented subspace

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,´ Uniërsidad de Extremadura, Aënida de Eläs, 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 complemented 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 homomorphism 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. Since the next Section 3 is independent on measure theory, the assumption on ␮ made in Theorem 1 turns out to be superfluous. We do not give the details here. NOTE 649

3. AVERAGING

In this section we shall see that, under rather mild assumptions on the

ground algebra, LAŽ.X1,..., Xn is a complemented subspace of L Ž.X1,..., Xn . Recall that an element u of a Banach algebra A is said to be unitary if 55u s 5uy1 5 s 1. The set of all unitary elements of A is a group under multiplication and will be denoted by U. ¨ ¨ THEOREM 2. Let Xi be Banach modules o er the commutati e Banach algebra A. If U spans a dense subspace of A, then LAŽ.X1,..., Xn is the ¨ range of a contracti e projection on L Ž.X1,..., Xn . ␮ Proof. Let d Ž.u1,...,uny1 be an invariant mean for the locally compact group U ny1 viewed as a discrete space.Ž We refer the reader tow 6, Chap. IVxwx or to the Greenleaf booklet 4 for information on invariant g means..Ž. For f L X1,..., Xn , put

PfŽ. x1 ,..., xn

s y1 иии y1 ␮ H fuxŽ.11,...,uxny1 ny11, u uny1 xdn Ž.u1 ,...,uny1 . U ny1 = иии = 5555F Clearly, Pf is a multilinear form on X1 Xn, with Pf f .On s g the other hand, it is clear that Pf f for all f LAŽ.X1,..., Xn and also that Pf depends linearly on f. Thus, the proof will be complete if we see that Pf is balanced for all f. A moment of reflection shows that it suffices to verify the identity s PfŽ.Ž. x1 ,...,uxin,..., x Pf x1 ,..., xin,...,ux F - g g for each 1 i n and all xkkX , u U. Using the invariance of the ␮ ¨ s mean d Ž.u1,...,uny1 and letting iiuu, we get

PfŽ. x1 ,...,uxin,..., x

s y1 иии y1 HfuxŽ.11,...,uuxii,...,ux ny1 ny11, u uny1 xn = ␮ d Ž.u1 ,...,uny1

s ¨ y1 иии ¨y1 y1 HfuxŽ.11,..., iix ,...,ux ny1 ny11, u u inu y1 xn = ␮ ¨ d Ž.u1 ,..., in,...,u y1

s y1 иии y1 HfuxŽ.11,...,uxii,...,ux ny1 ny11, u uny1uxn = ␮ d Ž.u1 ,...,uin,...,u y1 s PfŽ. x1 ,..., xin,...,ux , as desired. 650 NOTE

Theorem 1 now follows from Corollary 2Ž. and its Remark and Theorem

2. Taking into account that balanced forms on Lp spaces are automatically symmetric, we can adhere the following corollary to Theorem 1. r q r s F F ϱ COROLLARY 3. Suppose n p 1 q 1, with 1 p, q . Then Lq is a complemented subspace of the space of n-homogeneous polynomials on Lp. Remark 2. The hypothesis about A appearing in Theorem 2 can be understood as a stronger form of amenability.Ž Amenability is a central theme in the homology of Banach algebras; seewx 5 .. It seems very likely that LAŽ.X1,..., Xn is complemented in L Ž.X1,..., Xn provided A is an amenable algebra. This is true, for instance, for group algebras. It is worth noting that if LAŽ.X1,..., Xn is complemented in L Ž.X1,..., Xn by any bounded projection and A is amenable, then there is an A-module projection from L Ž.Ž.X1,..., XnAonto L X1,..., Xn . This is so because LAŽ.X1,..., Xn is a dual module over A and all dual modules over an amenable algebra are injective; seewx 5, Theorem VII.1.6.I .

4. DIAGONAL SUBSPACES OF TENSOR PRODUCTS

We close the paper by showing that Theorem 1 can be predualized for sequence spaces. k ϱ F F Let Xkiibe Banach spaces with bases Že . s1 for 1 k n. The main s m иии m ⌬ diagonal of X X1 ˆˆXn is the closed subspace spanned in X by m иии m ⌬ the diagonal vectors eiie . Let us show that is complemented in X provided each factor Xk has an unconditional basis.

LEMMA 2. Let X1,..., Xbemn -dimensional Banach spaces with 1-un- k m F F conditional basesŽ eii. s1 for 1 k n. Then the ‘‘diagonal’’ projection Q ¨ m иии m gi en on X1 ˆˆXbyn

1 m иии m n s иии s 1 m иии m n s eiŽ1. eiŽ n. if iŽ.1 in Ž .; QeŽ.iŽ1. eiŽn. ½ 0 otherwise

¨ is contracti e. Consequently, if X1,..., Xn areŽ. infinite-dimensional Banach k ϱ spaces with unconditional basesŽ eii. s1, then the diagonal projection is m иии m bounded on the space X1 ˆˆXn.

m Proof. We only prove the first part. Consider each Xk as an lϱ -module in the obvious way and let U be the unitary group of the Banach algebra m lϱ , endowed with the norm-topology. Obviously U is compact and so there NOTE 651 is a unique normalized Haar measure du on U. define a linear operator R m иии m on X1 ˆˆXn by

m иии m RxŽ.1 xn

s иии m иии m m y1 y1 HHux11 uxny1 ny11Ž.u ...uny1 xdun 1 ...duny1 .

Clearly, R is linear and bounded, with 55R F 1. Routine computations now show that R s Q. Hence Q is contractive too and the proof is complete. This lemma, in combination with Corollary 2, provides us with a very simple proof of the following nice result of Arias and Farmerwx 1 . From s ϱ now on, we make the convention that l p means c0 if p . m иии m PROPOSITION 1Ž. Arias and Farmer . The main diagonal of l ppˆˆl ¨ 1 r sn is a complemented subspace isomorphic to l p, where p is gi en by 1 p n r minÄ 1, Ýis1 1 pi4. ⌬ s m иии m Proof. We already know that is complemented in X l ppˆˆl m иии m ϱ 1 n by a contractive projection. We prove that Ž.eiiie s1 is an l p-basis. Take

m s m иии m x Ý xeiiŽ.e i. is1

r q иии q r F We shall consider two cases. First, suppose 1 p1 1 pn 1. Let s ª P : L Ž.1pp ,...,l X* L lpp Ž.l ,...,l be the averaging projection de- 1 n ϱ 1 n U scribed in the proof of Theorem 2. Since L Ž.l ,...,l s l s l and, lpϱ 1 pn p q taking into account that ²:²:Pf, x s f, x for all f and x g ⌬ Žthis is obvious. , we have

55 x X s sup ²:f , x s sup ²Pf , x : 55f F1 55f F1 s supÄ4²:f , x : f is induced by a norm-one lq-sequence

1rq s sup fx: <

⌬ so that is isometric to l p. 652 NOTE

r q иии q r ) G Now suppose 1 p1 1 pnkk1. Choose s p in such a way that r q иии q r s 55F 55 F F 1 s1 1 sn 1. Since y spkky for each 1 k n and all y g l the formal identity l mˆˆиии m l ª l m ˆˆиии m l has norm one. pppssk 1 n 1 n Hence,

~~m иии m and Ž.eiie is an l1-basis. This completes the proof.~~

~~5. CONCLUDING REMARKS~~

Perhaps a few remarks about the module-theoretic content of this note are in order. First, there is a classical construction which linearizes balanced forms. Indeed, if the Xi are modules over the commutative s m иии m Banach algebra A, then LAŽ.Ž.X1,..., Xn X1 ˆˆAAnX * in a canon- m ical way. In fact, this identity could be taken as the definition of X1 ˆ A иии m wx ˆ AnX in 5 . To obtain a suitable description of the tensor product of modules on an algebra, consider the usualŽ. projective, Banach tensor m иии m product X1 ˆˆXn and the closed subspace N spanned by the elements of the form

~~m иии m m иии m m иии m Ž.x1 axijnx x y m иии m m иии m m иии m Ž.x1 xijnax x .~~

m иии m m иии m r Then X1 ˆˆAAnX equals Ž.X1 ˆˆXn N. Thus, Corollary 2 and the comments made after Theorem 1 imply that if r q иии q r F mˆˆиии m s Ž. 1 p1 1 pnpLLpp1, then L ϱϱL L as Lϱ-modules , r s r q1 иии q r n ␮ where p is given by 1 p 1 p1 1 pn. Surprisingly enough, if is nonatomic and 1rp q иии q1rp ) 1, then L mˆˆиии m L s 0 since 1 npLLp1 ϱϱn in this case the space of balanced forms reduces to zero. We remark, however, that this situation is not too strange in the pure algebraic setting: it is easily checked that ޑrޚ mޚ ޑrޚ s 0.

~~ACKNOWLEDGMENTS~~

~~It is a pleasure to thank Ricardo Garcıa´´ Gonzalez and Pedro Sancho de Salas for several valuable remarks which where obvious to them, but not to me. NOTE 653~~

~~REFERENCES~~

1. A. Arias and J. D. Farmer, On the structure of tensor products of l p-spaces, Pacific J. Math. 175 Ž.1996 , 13᎐37. 2. M. Gonzalez,´ R. Gonzalo, and J. Jaramillo, Symmetric polynomials on rearrangement-invariant function spaces, J. London Math. Soc. Ž.2 59 Ž1999 . , 681᎐697. 3. R. Gonzalo, Multilinear forms, subsymmetric polynomials and spreading models on Banach spaces, J. Math. Anal. Appl. 202 Ž.1996 , 379᎐397. 4. F. P. Greenleaf, ‘‘Invariant Means on Topological Groups and Their Applications,’’ Van Nostrand, Princeton, NJ, 1969. 5. A. Ya. Helemskii, ‘‘Banach and Locally Convex Algebras,’’ Oxford Science Publications, 1993. 6. E. Hewitt and K. A. Ross, ‘‘Abstract Harmonic Analysis, I,’’ Springer-Verlag, New YorkrBerlin, 1979. 7. J. Lindenstrauss and L. Tzafriri, ‘‘Classical Banach Spaces, II,’’ Springer-Verlag, New YorkrBerlin, 1979.