RENDICONTI DEL CIRCOLO MATI';MATICO DI PALERMO Serie 11, Tomo XLVII (1998), pp. 493-502

ABOUT T H E P R O P E R T I E S (V) AND (R.D.P.) IN I N J E C T I V E T E N S O R P R O D U C T S RAFFAELLA CILIA

In questo lavoro vogliamo provare che se X ~ uno spazio di Banach del tipo Leo con la proprietor (V) of Pelczynski (rispettivamente con la Reciprocn Propriet~ di Dunford Pettis) c Y b uno spazio di Banach riflessivo allora X | Y ha la propriet/~ (V) (rispettivamente la Reciproca ProprietS. di Dunford Pettis).

In this paper we want to study some properties of the injective tensor product of two Banach spaces (for the definition of this space one can see [3]). Our target is to find conditions on the Banach spaces X and Y such that the injective tensor product X | Y verifies the property (V) of Pelczynski or the Reciprocal Dunford Pettis property (R.D.P.). We recall that a Banach space X has the property (V) of Pelczynski if every unconditionally converging operator ([4]) defined on it with values in a Banach space is weakly compact; X verifies the Reciprocal Dunford Pettis property if every Dunford Pettis operator on it ([4]) is weakly compact. From the previous definitions one can easily see that every space with the property (V) has A.M.S. classification: 46M05. Work partially supported by M.U.R.S.T. (40%). Key words: Injective tensor product, Property (V), Reciprocal Dunford Pettis Property.

494

RAFFAELLA CILIA

the (R.D.E) property. E Cembranos, N. Kalton , E. Saab and E Saab ([2]) have studied the problem of finding hypotheses on the Banach space Y such that the space C(K, Y) ~- C(K) | Y of the Y valued continuous functions on a compact space K has the property (V). Now we want consider injective tensor products X | Y where X belongs to a class of spaces more large than the class of C(K) spaces. This is the class of Loo spaces (see ([1] for the definition) In the present note we want to assure that if X is a Loo space with the property (V) (respectively with the (R.D.E) property) and Y is a reflexive Banach space then X | Y verifies the property (V) respectively the (R.D.E) property). Next, as a corollary, we prove that if X is a Loo space with the (R.D.E) property and Y is a Banach space not containing a complemented copy of Ii and with an unconditional Schauder decomposition (Yn),, such that every subspace Yn is reflexive, then X | Y verifies the (R.D.E) property. In order to prove our target we shall use some results concerning with the Loo spaces. It is well known that if X is a L ~ space, then X* is isomorphic to a closed subspace of the space M ( K ) of the regutar scalar Borel measures of bounded variation defined on the compact space K = (Bx***, w*). We recall the construction of this isomorphism. Since X is a Loo space then there exists a projection P from the space C(K) onto the bidual space X** where K is the unit ball Bx.,, endowed with the w* topology ([1]). Now we consider the adjoint operator

P* 9 X*** --~ C(K)* Let s be the restriction of P* to the subspace X*. One can easily verify that s is an isomorphism of X* onto a closed subspace of (C(K))* ~- M(K). Indeed

Ils(x*)ll(C(K))* = sup{ls(x*)(~b)l : ~ ~ C(K), IIq)l[ < 1} = = sup{Ix*(P(~b))] " q~ ~ C(K), 114)115 1} 5 ]lx*ll IIPll. On the other hand, since X** is a subspace of C(K), we have

lls(x*)ll(c(K)), >__sup{ts(x*)(x**)l" x

X**, llx**ll <_ 111 = iix*lt

Then we have

IIx*lt _< IIs(x*)ll _< IIx*ll IIPII.

ABOUT THE PROPERTIES

(V)

AND (R.D.P.)

IN INJE(TrlVE

TENSOR

PRODUCTS

495

In what follows we need some information on the dual of the injective tensor product space X | Y in terms of vector measures. So we prove the following LEMMA. Let X be a Loo space, then there exists an isomorphism of (X@e Y)* onto a subspace of the space M ( K , Y*) of all regular Y* valued measures with bounded variation defined on the compact space K = (Bx***, w*). Proof Let P 9 C(K) ~ observe that the operator

X** the above projection. Now we

9 C ( K ) @~ Y ~- C ( K , Y) --+ X** |

Y

defined by /5.(r174174

vCEC(K),

y~ Y

is a proiection since X** @~ Y is a subspace of C ( K ) @e Y and obviously /5 on it is the identity operator. Let

P* 9 (X** |

Y)* --> (C(K, Y))* _~ M(K, Y*)

be the adjoint operator of /5. It is well known (see [3]) that (X | Y)* is isometrically isomorphic with the space B ^ ( X , Y) of the integral bilinear continuous forms on X x Y. Since the natural inclusion of B ^ ( X , Y) into B ^ ( X **, Y) is an isometry (see [3]), it follows that (X| Y)* is a closed subspace of the space B ^ ( X **, Y) ~(X** | Y)*. So it is possible to consider the restriction S of /5* to (X @e Y)*. We prove that S is an isomorphism of (X @~ Y)* onto a subspace of (C(K, Y))*. Let L be an arbitrary element of (X@~Y)*. Then we have IIS(L)II(Cr

= sup{lS(L)(~)l " ap ~ c ( g , Y), II'Pll _< 1} = = sup{IL(/5(gz))l 9 ~ ~ C ( K , Y), 117zll _< 1} < IILII IIPll.

On the other hand, since X**|

is a subspace of C ( K , Y), one

496

RAFFAF.LLA

(;ILIA

has

IIS(L)IIr162

~ sup{IS(L)(~)l 9 ~ ~ X** | Y, I1~11 ~ 1}

=

= sup{lL(~)l 9 ~ ~ X** | Y, I1~11 5 1} = IILII Then we have IILI[ _< IIS(L)II _< IILII IlPll. Using the previous lemma we can prove the following THEOREM 1. Let X be a L ~ space with the property (V) oJ" Pelczynski. Let Y be a reflexive Banach space. Then X @~. Y has the property (V) too.

Proof. Let T : X @~ Y --+ Z be an unconditionally converging operator. Now for every choice of y in By and z* in Bz. we consider the functional (y, T'z*) on X defined by putting (y, T*z*)(x) = (T*z*)(x | y) = (z*, T(x | y)). Obviously this functional is linear and continuous; then the set

H = {(y, T'z*) : IIYll -< 1, IIz*ll _< l} is contained in X*. We want to prove that H is weakly relatively compact in X*. In order to do that, since X has the property (V), it is enough to prove that

limsup{l(h,x,,)l : h ~ H} = 0 ll ('20,

for any weakly unconditionally converging series ~-~x,, in X, that ,1=1 is limsup{l(y, T*z*)(xn)l : IlYll _< 1, IIz*ll _< 1) = 0 n o(3

for any weakly unconditionally converging series ~ x n Ilz|

Since for any y 6 Y one has sup{l(z*, T(x~ | Y))I : llz*ll _< 1} = IIT(x, | y)ll

in X.

ABOUT TItE PROPERTIES

(V)

AND (R,D,P.)

IN INJECTIVE TENSOR

497

PRODUCTS

it is enough to prove that for every weakly unconditionally converoo

ging series ~-~'x. in X one has n=l

limsup{llT(x~ |

: Y c Y, [lYll < I} = 0.

n

If this were not true, then there would exist a positive number OO 8, a subsequence (x',,)~= 1 of (X n).=l and a sequence (Yn)~=l of elements in the unit ball of Y such that

llT(x'. | Y)II > ~

(*)

Vn e N

oo

The series Z

x'. | y. is, as one can easily prove, weakly un-

n=l

conditionally converging in X |

Y thanks to the facts that the ele0(3

ments y. have norm equal or less than one and the series y ' ~ x 'n n=l

is weakly unconditionally converging. From (*) it follows that the oo

series ~ _ T ( x ' n | y.) cannot converge in the norm of Z and this q

n=l

contradicts the fact that T is unconditionally converging. So we have proved that H is relatively weakly compact. Let s and S be the isomorphisms whose existence is proved in the previous lemma. So for any y ~ By and any z* ~ Bz*, we can consider the scalar Borel regular measure s((y, T'z*)) that corresponds with the functional (y, T'z*). At the same time we can look at the scalar Borel regular measure (y, S(T*z*)) defined as a functional on C(K) by the law

(y, S(T*z*)I(r

= (S(T* z*))(r | y)

V r ~ C(K).

Then we want to prove that

(y, S(T* z*))(4;) = s(ly, T*z*))(qS)

u

~ C(K).

Let 4' be fixed in C(K). By the construction of the isomorphism S it follows that

(y, S(T*z*))(dp) : (S(T*z*))(r | y) = = (T*z*)(P(r | y)) = (T*z*)(P(r | y) = : (y, T*z*)(P(r

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RAFFAELLA CILIA

On the other hand by the construction of s it results

s((y, T*z*))(~p) = (y, T*z*)(P(dp)). Then we can say that

s(H) = {(y, S(T*z*)) : IlYll _< 1, IIz*ll _< 1}. Since H is weakly relatively compact and s is an isomorphism,

s(H) is also weakly relatively compact. So {(y, S(T*z*)):llYll _< 1, IIz*ll _< 1} is weakly relatively compact in M ( K ) and then it is uniformly countable additive ([3]). It follows that the set of the variations W = {I(Y, S(T*z*))I : IlYll _< 1, IIz*ll _< 1} is uniformly countable additive too ([3]). Since Y is reflexive this is sufficient to say that the set

{S(T*z*) : IIz*II < 1} is also uniformly countable additive ([3]). Thanks to another well known result ([3]) the set of the total variations

{IS(T*z*)l : llz*ll <_ 1} is uniformly countable additive. Now we observe that from the proof of the work of Pelczynski ([7]) concerning with the property (V) in the space C ( K , Y) it follows that if Y is reflexive, a subspace H of (C(K, Y))* is weakly relatively compact if and only if the set of the total variations of the elements in H is uniformly countable additive. Then we can say that the set

{S(T*z*) : llz*ll < 1} is weakly compact. Since S is an isomorphism the set

{T*z* : llz*ll <_ l} is weakly compact too. So T* is weakly compact and hence so is T.

Remark. With the same proof as in Theorem 1 one can generalize the result in ([9]) assuring that if X is a L ~ space with property

ABOUT THE PROPERTIES

(V)

AND ( R . D . P . )

IN INJECTIVE TENSOR

PRODUCTS

499

(V) and Y is a Banach space whose dual does not contain a copy of Ii then an operator T on X| Y is unconditionally converging if and only if its adjoint is weakly precompact. Now we prove a theorem analogous to the previous one for the (R.D.P.) property. THEOREM 2. Let X be a Loo space with the Reciprocal Dunford Pettis property. Let Y be a reflexive Banach space. Then X | Y has the Reciprocal Dunford Pettis property too.

Proof It is enough to repeat the same proof as in Theorem 1, observing that in this case the weak relative compactness of the subset H follows from a characterization of the spaces with the Reciprocal Dunford Pettis property due to T. Leavelle ([6]). It assures that a Banach space X has the Reciprocal Dunford Pettis property if and only if every set K in X* such that limsup{l(x*,x~)l : x* ~ K } = 0 n O0 for every weakly null sequence (X ,,)~=1 in X is weakly relatively compact. We want to improve the hypothesis on Y in Theorem 2 with the following.

COROLLARY. Let X be a Loo space with the Reciprocal Dunford Pettis property. Let Y be a Banach space with no complemented copy of ll and with an unconditional Schauder decomposition (Yn)n where every Yn is reflexive. Then X | Y has the Reciprocal Dunford Pettis property.

Proof By Theorem 2 it follows that every space X | Y,, has the Reciprocal Dunford Pettis property. So in order to prove our result it is enough to assure that (X | Yn)n is a shrinking Schauder decomposition for X | Y (see [5]). It is very easy to verify that (X | Y,,)~ is a Schauder decomposition for X | Y. It is enough to

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define for each n the projection ~,~ 9 X |

Xi ~ Yi

Y ---> X |

Yn by

Xi ~ Vn(Yi)

:

i =1

where (v,,)~= l is the system of projections associated with the Schauder decomposition (Yn)~ of Y. Thanks to a well known result (see [10]), we must only prove that (~*(X | Y)*)~=l is a Schauder decomposition for (X | Y)*. We have to show that for every element oo

L in the space ( X |

Y)* the series Z~3,~(L) converges in the norm n=l

of (X|

to L. Let L be a fixed element in (X| OO

that the series Z

We observe

fi,~(L) is w*-unconditionally converging that is

n=l Oo

I~*(L)(u)l

< +e~

for every u in X | Y. Fix arbitrarily u; one can suppose that u -r

)__sxj |

yj. It results

j=l

P

S,

n-~-q

)l

P

:

n=q

L

xj | vn(yj)

n=q r

p

j=l

n=q

~--~Tt.(xjl(v,,(yj)) n=q

j=l

where TL is the integral operator from X into Y* associated with the oo

element L. For each j < r we look at the series Z

ITL(xj)(vn(yj))l.

n=l

co

Since (Yn)n is an unconditional decomposition, the series ~_vn(yj) n=l

is unconditionally converging and so it is also weakly unconditionally

ABOUT THE PROPERTIES (V) AND (R.D.P.)

IN INJECTIVE TENSOR PRODUCTS

50l

oo

converging. It follows that the series ~

ITL(xj)(vn(yj))l

is conver-

n=l

gent. Fixed e > O, one can find an index vj such that if p > q > vj then P

~ IrL(xj)(vn(yj))l < e/r. n=q

If v = m a x v j J

and p > q >

v then

P

I ~ * ( L ) ( u ) l < ~. n=q

With the w*-unconditional convergence in hand, by a known reO0

sult ([4], pag. 49), it follows that the series ~

fi~*(L) is unconditio-

n=l

nally converging if and only if the space (X | Y)* does not contain an isomorphic copy of loo. Now we recall that, by our lemma, (X | Y)* is a subspace of (C(K, Y'))* where K is a suitable compact space. By our hypothesis Y does not contain a complemented copy of 11 and then C ( K , Y) can not contain a complemented copy of 11 ([8]). By the famous Theorem of Bessaga and Pelczynski it follows that ( C ( K , Y))* and so (X | Y)* does not contain loo. This oo

means that ~

~*(L) converges unconditionally. Since one can easily

n=l oo

prove that in the w* topology ~

fin(L) converges to L, L must be

n=l

its limit in the norm topology of (X |

Y)*. So we are done.

REFERENCES [1] Bourgain J., New classes of LP spaces, Springer Verlag, 1981. [2] Cembranos P., Kalton N.J., Saab E., Saab P., Pelczynski's Property (V) on C(g2, E) Spaces, Math. Ann. 271, 91-97, 1985. [3] Diestel J., Uhl J.J., Vector Measures, Math. Surveys, 15, 1977. [4] Diestel J., Sequences and Series in Banach Spaces, Springer Verlag, 1984.

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[5] Emmanuele G., Property (V) and some of its relatives, to appear [6] Leavelle T., The Reciprocal Dunford Pettis Property, to appear in Ann. Mat. Pura ed Applicata. [7] Pelczynski A., Banach Spaces on which ever), unconditionally converging operator is weakly compact, Bull. Acad. Polon. Sci. vol. 10, No. 12 (1962), 641-648. [8] Saab E., Saab P., A stability property, of a class of Banach spaces not containing a complemented copy of Ii, Proc. Arner. Math. Soc., 84 (1982), 44-46. [9] Saab E., Saab E, On unconditionally converging and weakly precompact operators, I!1. Journal of Math., 35, n. 3 (1991), 522-531. [10] Singer I., Bases in Banach Spaces, Springer Verlag, vol. 2, 1981. Pervenuto il 12 maggio 1997.

Department of Mathematics University of Basilicata Potenza Via Nazario Sauro 85 85100 Potenza