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J. Math. Anal. Appl. 315 (2006) 544–554

www.elsevier.com/locate/jma

Relatively weakly open subsets of the unit ballin functions spaces

Julio Becerra Guerrero, Ginés López Pérez∗

Universidad de Granada, Facultad de Ciencias, Departamento de Matemática Aplicada, 18071 Granada

Received 17 June 2004

Available online 5 August 2005

Submitted by R.M. Aron

Abstract

For an infinite Hausdorff compact setK and for any Banach spaceX we show that every nonempweak open subset relative to the unit ball of the space ofX-valued functions that are continuous whX is equipped with the weak (respectively norm, weak-∗) topology has diameter 2. As consequenwe improve known results about nonexistence of denting points in these spaces. Also we charwhen every nonempty weak open subset relative to the unit ball has diameter 2, for the spBochner integrable and essentially bounded measurableX-valued functions. 2005 Elsevier Inc. All rights reserved.

Keywords:Denting point; Slices; Weak open subsets

1. Introduction

The nonexistence of denting points in the unit ball of some functions spaces has bsubject of several recent researching [8,14]. A pointx0 in the sphere of a Banach spaceX,SX , is a denting point of the unit ball inX, BX , if there are slices, that is, subsets inway

S(x∗, α) = x ∈ BX: x∗(x) > ‖x∗‖ − α, x∗ ∈ X∗, α ∈ R

,

* Corresponding author.E-mail addresses:juliobg@ugr.es (J. Becerra Guerrero), glopezp@ugr.es (G. López Pérez).

0022-247X/$ – see front matter 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.jmaa.2005.06.063

J. Becerra Guerrero, G. López Pérez / J. Math. Anal. Appl. 315 (2006) 544–554 545

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, thef.it ballall has].

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containingx0, with diameter arbitrarily small. From [11],x0 is a denting point of the unball of X if and only if x0 is an extreme point inBX and x0 is a point of weak-normcontinuity, that is, a point of continuity for the identity map from(BX,w) onto (BX,n),wherew and n denote the weak and the norm topology, respectively. In particularexistence of denting points in the unit ball of a Banach spaceX implies the existence ononempty weak open subsets relative to the unit ball inX with diameter arbitrarily smallThen the extremely opposite property to the existence of denting points in the unof a Banach space is that every nonempty weak open subset relative to the unit bdiameter 2. This is the case, for example, forC∗-algebras [2], and uniform algebras [13The aim of this note is to show the existence of functions and operator spaces whernonempty weak open subset relative to the unit ball has diameter 2, by improvinresults about nonexistence of denting points in [8] and [14].

Given a Banach spaceX and a Hausdorff compact topological spaceK , we noteC(K,X) the Banach space of all continuous functions onK into X with the norm topol-ogy,WC(K,X) the Banach space of all continuous functions onK into X with the weaktopology andW ∗C(K,X) the Banach space of all continuous functions onK into X withthe weak-∗ topology, whenX is a dual space. All these spaces are equipped with the smum norm.

The first goal of this note is to prove that every nonempty weak open subset reto the unit ball ofC(K,X), WC(K,X) andW ∗C(K,X∗) has diameter 2 wheneverK beinfinite (Corollary 2.4). This result follows from the study of the spaceC(K, (X, τ)) ofcontinuous functions on a Hausdorff compact topological spaceK into a Banach spacX endowed with a Hausdorff locally convex topologyτ (Theorem 2.3). We will obtainsome consequences from this result, by using well-known identifications for this sFor example, every nonempty weak open subset relative to the unit ball inL(X,C(K))

(the linear and continuous operators space) has diameter 2, wheneverK be infinite (Corol-lary 2.6). Turning to injective tensor products, we show that for any infinite-dimensspaceY such thatY ∗ is isometric to anL1(µ)-space (the so-calledL1 predual spaces [9]and for any Banach spaceX, every nonempty weak open subset relative to the unitof X ⊗ε Y has diameter 2 (Corollary 2.9). As a consequence, we obtain that the spbilinear forms onX × Y is extremely rough, wheneverY be an infinite-dimensionalL1

predual.Also, we characterize when every nonempty weak open subset relative to th

ball has diameter 2 for the operators spaceL(L1(µ),X), Theorem 2.11, and for thspacesL1(µ,X), of Bochner integrableX-valued functions andL∞(µ,X), of essentiallybounded and measurableX-valued functions, whereX is a Banach space andµ is a posi-tive measure, Theorem 2.13.

2. The main result

We begin with two elementary lemmas which will be essential for the main results

Lemma 2.1. LetX,Y Banach spaces andZ = X ⊕1 Y .

546 J. Becerra Guerrero, G. López Pérez / J. Math. Anal. Appl. 315 (2006) 544–554

.

e

,

bi-eak

ubset

(i) Assume thatX has a nonempty weak open subset relative toBX with diameter lessthan2. ThenZ has it too.

(ii) Assume that every nonempty weak open subset relative toBX or BY has diameter2.Then every nonempty weak open subset relative toBZ has also diameter2.

Proof. (i) Let U be a nonempty weak open subset relative toBX with diameter less than 2ThenU must contain a nonempty finite intersection of slicesW = x ∈ BX: x∗

i (x) > 1−α,1 i p, where 0< α < 1, p is a natural number, andx∗

i ∈ SX∗ for all 1 i p.Consider nowV = (x, y) ∈ BZ: x∗

i (x) > 1 − α, 1 i p. HenceV is a nonemptyweak open subset relative toBZ . Pick (x, y), (x′, y′) ∈ V . Taking into account that thdiameter ofU is less than 2 and‖y − y′‖ < 2α, we deduce that∥∥(x, y) − (x′, y′)

∥∥ = ‖x − x′‖ + ‖y − y′‖ diam(U) + 2α.

Finally, it is enough choseα small to conclude that the diameter ofV is less than 2.(ii) Let U be a nonempty weak open subset relative toBZ . Then, by [7, Lemma II.1]

there is a convex combination of slices∑n

i=1 λiSi contained intoU , wheren is a naturalnumber,λi1in are positive real numbers with

∑ni=1 λi = 1, and for every 1 i n,

we put

Si = (x, y) ∈ BZ: x∗

i (x) + y∗i (y) > 1− αi

,

where(x∗i , y∗

i ) ∈ SZ∗ and 0< αi < 1.Now, we split the set1, . . . , n in two disjoint subsetsI , J , such that‖x∗

i ‖ = 1 foreveryi ∈ I and‖y∗

j ‖ = 1 for everyj ∈ J .For everyi ∈ I we putTi = x ∈ BX: x∗

i (x) > 1 − αi and for everyj ∈ J , we putRj = y ∈ BY : y∗

j (y) > 1− αj . ThenTi is a family of slices inBX andRj is a familyof slices inBY , satisfying that(Ti,0) ⊂ Si and (0,Rj ) ⊂ Sj for every i ∈ I andj ∈ J ,

respectively. Thus, it follows that(∑

i∈Iλi

λITi,0) ⊂ ∑

i∈Iλi

λISi and (

∑j∈J

λj

λJRj ,0) ⊂∑

j∈J

λj

λJSj , whereλI = ∑

i∈I λi andλJ = ∑j∈J λj .

The sets∑

i∈Iλi

λITi and

∑j∈J

λj

λJRj have diameter 2, since they are convex com

nation of slices inBX and BY , respectively, and, by hypothesis, every nonempty wopen subset relative toBX , respectivelyBY , has diameter 2. Hence, givenε > 0, there arex, x′ ∈ ∑

i∈Iλi

λITi andy, y′ ∈ ∑

j∈J

λj

λJRj verifying ‖x − x′‖,‖y − y′‖ > 2− ε.

Note that(λI x,λJ y), (λI x′, λJ y′) ∈ ∑n

i=1 λiSi and so

diam(U) diam

(n∑

i=1

λiSi

) λI‖x − x′‖ + λJ ‖y − y′‖ > 2− ε.

Lemma 2.2. Let X be a Banach space satisfying that every nonempty weak open srelative toBX has diameter2. Then every nonempty weak open subset relative toBX⊕∞Y

has diameter2, whereY is an arbitrary Banach space.

Proof. We callZ = X ⊕∞ Y and letP : Z → X be the projection fromZ ontoX, whichis weak open. It is clear thatBZ = BX × BY and‖P ‖ = 1. Then ifW is a weak open inZ

J. Becerra Guerrero, G. López Pérez / J. Math. Anal. Appl. 315 (2006) 544–554 547

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ce

tive toral,

ir

ryp

.

o-

ly

t

-sor

such thatW ∩ BZ = ∅, one has thatV = P(W ∩ BZ) is a nonempty weak open relativeBX and, so diam(V ) = 2. Hence diam(W ∩ BZ) = 2.

In order to show our main result we begin with some notation. For a Banach spaX,a Hausdorff compact topological spaceK , and a Hausdorff locally convex topologyτon X, C(K, (X, τ)) stands for the vector space of all continuous functions fromK into(X, τ). The main problem in order to study the diameter of weak open subsets relathe unit ball ofC(K, (X, τ)) is to have the possibility of consider the sup norm. In genethis is not possible. A natural condition onτ for this is assume thatτ is compatible forthe dual pair(X,X∗), since Mackey’s Theorem states that theτ -bounded subsets ofXare also bounded for the norm topology onX. Recall thatτ is compatible for the dual pa(X,X∗) if (X, τ)∗ = X∗ and that every compatible topology onX for the dual pair(X,X∗)is coarser than the norm topology onX, by Mackey–Arens Theorem (for duality theoresults we reference to [12,15]). So, we have thatC(K, (X, τ)) endowed with the sunorm is a Banach space wheneverτ is a compatible topology for the dual pair(X,X∗) andmoreover everyτ -neighborhood onX is also a neighborhood onX for the norm topologyNow, with this facts in mind, we present our main result.

Theorem 2.3. Let X be a Banach space and letK be an infinite Hausdorff compact toplogical space. LetZ = C(K, (X, τ)) the space of continuous functions fromK to (X, τ),whereτ is a Hausdorff locally convex topology onX. Assume thatτ is compatible for thedual pair (X,X∗). Then every nonempty weakly open set relative to the unit ball ofZ hasdiameter2. Moreover, ifX is a dual space with predualX∗, then every nonempty weakopen set relative to the unit ball ofZ has diameter2, wheneverτ be compatible for thedual pair (X∗,X).

Proof. PickW a weak neighborhood ofZ and separate the proof in three steps.

(1) Assume thatK have many infinite isolated points. Taketn a sequence of differenisolated points ofK andt0 an accumulation point oftn. Now we have that there isa ∈SZ ∩ W such thata(t0) = 0, sinceSZ ∩ W has nonempty interior relative to the sphereSZ ,for the norm topology. Chose aτ -neighborhoodU of 0 in X and aτ -neighborhoodV ofa(t0) in X, such thatU ∩ V = ∅.

By the continuity ofa, we know that there isI an infinite set of positive integers numbers such thata(tn): n ∈ I ⊂ V , and soa(tn): n ∈ I ∩ U = ∅. Therefore, there iδ > 0 such that‖a(tn)‖ > δ ∀n ∈ I , since everyτ -neighborhood is also neighborhood fthe norm topology inX. In the sequel, we assume without loss of generality thattn isa sequence of different isolated points ofK satisfying‖a(tn)‖ > δ ∀n, for suitableδ > 0.

For eachn we considerxn ∈ C(K) given by

xn(tn) = 1

‖a(tn)‖ , xn(tn+1) = − 1

‖a(tn+1)‖ , xn(t) = 1,

∀t ∈ K \ tn, tn+1.Now, xn is a bounded sequence inC(K) satisfying limn xn(t) = 1 for eacht ∈ K .

Applying [4, Theorem VII.1], we obtain thatxn converges to1 in the weak topology

548 J. Becerra Guerrero, G. López Pérez / J. Math. Anal. Appl. 315 (2006) 544–554

lt

f

d

l

t

anachace

l-

al

in C(K), where1 is the constant function inC(K) equal 1. Then the sequencexnaconverges toa in the weak topology ofZ, sincea ∈ SZ . Hence there isp such thatxpa, xp+1a ∈ W ∩ BZ , sincexna ∈ BZ ∀n.

Finally,

diam(W ∩ BZ) ‖xn+1a − xna‖ ∣∣xn+1(tn+1) − xn(tn+1)

∣∣∥∥a(tn+1)∥∥ = 2.

(2) Assume that there isa ∈ W ∩ SZ such that for allδ > 0 there ist0 ∈ K ′ verifying‖a(t0)‖ > 1− δ. (K ′ denotes the set of accumulation points ofK .)

For eachδ > 0 we considerW0 = t ∈ K: f (a(t)) > 1 − δ, wheref ∈ SX∗ satisfiesf (a(t0)) > 1 − δ. The existence off is guaranteed becauseτ is compatible for the duapair (X,X∗) (note that in the caseX be a dual space with predualX∗, the same argumenworks considering the dual pair(X∗,X)).

Now, W0 is an open subset ofK containingt0. Hence there isWn a sequence ononempty open subset pairwise disjoint ofW0. For eachn take tn ∈ Wn andxn ∈ C(K)

such thatxn(tn) = −1 andxn(t) = 1 whenevert ∈ K \ Wn. From [4, Theorem VII.1]xn

converges weakly to1 in the weak topology ofC(K). Thenxna converges weakly toa inthe weak topology ofZ. Thus there isn such thatxna ∈ BZ ∩ W and

‖a − xna‖ 2∥∥a(tn)

∥∥ 2f(a(tn)

) 2(1− δ).

Finally diam(W ∩ BZ) = 2, sinceδ was arbitrary.

(3) Assume that the set of isolated points ofK is finite. Then we can write:

Z = n∞(X) ⊕∞ C(K, (X, τ)

),

whereK is a perfect compact Hausdorff topological space andn is the number of isolatepoints inK . By (2) and Lemma 2.2, the proof is complete.

As the weak and the norm topologies on a Banach spaceX are compatible for the duapair (X,X∗) and the weak-∗ topology onX∗ is compatible for the dual pair(X∗,X), weobtain the following consequence.

Corollary 2.4. Let X be a Banach space and letK be an infinite Hausdorff compactopological space. Then every nonempty weakly open relative to the unit ball ofC(K,X),WC(K,X) or W ∗C(K,X∗) has diameter2.

We want remark that the above result can be proven, in a similar way, for the Bspacec0(L,X) of continuous functions on a Hausdorff locally compact topological spL which vanish at infinite with values in a Banach spaceX equipped with the norm topoogy, where the spacec0(L,X) is endowed with the sup norm.

Corollary 2.5. Let X be a Banach space and letK be an Hausdorff compact topologicspace. LetZ be one of the Banach spacesZ1 = C(K,X), Z2 = WC(K,X) and, ifX is adual space,Z3 = W ∗C(K,X). Then

J. Becerra Guerrero, G. López Pérez / J. Math. Anal. Appl. 315 (2006) 544–554 549

ball

the

e

ball

rm

f

at thehrsly.

al

ball

pacese

(i) Every nonempty weakly open relative to the unit ball ofZ has diameter2 if and onlyif K is infinite orK is finite and every nonempty weakly open relative to the unitof X has diameter2.

(ii) Z has denting points if and only ifK is finite andX has denting points.(iii) Z has points of weak-norm continuity if and only ifK is finite andX has points of

weak continuity.

Proof. (i) By Corollary 2.4, we know that every nonempty weakly open relative tounit ball of Z has diameter 2, wheneverK be infinite. Assume thatK is finite. ThenZ = n∞(X), wheren is the cardinality ofK . By Lemma 2.2, only it remains to decidthe caseK finite andX has a nonempty weak open subsetV relative to the unit ball withdiameter less than 2. Nown∞(V ) is a nonempty weak open subset relative to the unitof Z with diameter less than 2 and the proof is complete.

(ii) and (iii) By Corollary 2.4, we know thatZ has no denting (respectively weak-nocontinuity) points wheneverK be infinite. Assume thatK is finite. ThenZ = n∞(X),wheren is the cardinality ofK . In this case is straightforward to see that a pointz in BZ isa denting (respectively weak-norm continuity) point if and only if every coordinate oz isa denting (respectively weak-norm continuity) point inBX .

In order to obtain consequences of the above result, we recall (see [5]) thspacesC(K,X∗), WC(K,X∗) and W ∗C(K,X∗) can be isometrically identified witK(X,C(K)) (the compact operators space),F(X,C(K)) (the weakly compact operatospace) andL(X,C(K)) (the space of all linear and continuous operators), respective

Corollary 2.6. Let X be a Banach space and letK be an Hausdorff compact topologicspace. LetZ be one of the Banach spacesK(X,C(K)), F(X,C(K)) and L(X,C(K)).Then

(i) Every nonempty weakly open relative to the unit ball ofZ has diameter2 if and onlyif K is infinite orK is finite and every nonempty weakly open relative to the unitof X has diameter2.

(ii) Z has denting points if and only ifK is finite andX has denting points.(iii) Z has points of weak-norm continuity if and only ifK is finite andX has points of

weak continuity.

CallβN the Stone–Cech compactification ofN. Then it is straightforward thatL(1) andL(∞) can be isometrically identified withL(c0,C(βN)) andL(∞,C(βN)), respectively.So we obtain the following

Corollary 2.7. Every nonempty weakly open subset relative to the unit ball ofL(1) orL(∞) has diameter2.

The arguments used in the proof of Corollary 2.4 also work for the closed subsof L(X,C(K)) containingC(K,X∗) such that areC(K)-modules. This is the case for thspacesL(c0) andL(L1(µ)) and as a consequence we obtain the following

550 J. Becerra Guerrero, G. López Pérez / J. Math. Anal. Appl. 315 (2006) 544–554

open

o

,

ball inhen this

.ubsetl ofm

-

he unit

.ball

Corollary 2.8. Every nonempty weakly open subset relative to the unit ball ofL(c0) orL(L1(µ)) has diameter2.

In the following consequence, we apply our main result in deciding about weakrelative to the unit ball of injective tensor products spaces.

Corollary 2.9. Let X be a Banach space and letY be an infinite-dimensionalL1 predual.Then every nonempty weakly open subset relative to the unit ball ofX⊗ε Y has diameter2.

Proof. From the hypothesis, it is known thatY ∗∗ must be isometrically isomorphic tC(K) for suitable infinite Hausdorff compact topological space. Pick nowW a nonemptyopen subset ofX ⊗ε Y . Then, by the weak-star lower semicontinuity of the dual norm

diam(W ∩ BX⊗εY ) = diam(Ww∗ ∩ B(X⊗εY )∗∗

).

It is known [6] thatX⊗ε Y ∗∗ is a subspace of(X⊗ε Y )∗∗ containingX⊗ε Y , from one de-duce that diam(W ∩BX⊗εY ) = diam(Ww∗ ∩BX⊗εY ∗∗). Having into account thatX ⊗ε Y ∗∗is isometrically isomorphic toC(K,X) and Corollary 2.4 the proof is complete.

We note that the fact that every nonempty weak open subset relative to the unita Banach space has diameter 2 implies consequences in the dual and predual, wexists.

Let X be a Banach space. We recall that foru in SX, one define theroughness ofX at u,η(X,u), by the equality

η(X,u) := lim sup‖h‖→0

‖u + h‖ + ‖u − h‖ − 2

‖h‖ .

We remark that the absence of roughness ofX at u (i.e., η(X,u) = 0) is nothing but theFréchet differentiability of the norm ofX at u [3, Lemma I.1.3]. Givenε > 0, the BanachspaceX is said to beε-rough if, for everyu in SX, we haveη(X,u) ε. We say thatX isroughwhenever it isε-rough for someε > 0, andextremely roughwhenever it is 2-rough

Assume thatX is a Banach space satisfying that every nonempty weak open srelative to the unit ball inX has diameter 2. Then, by [3, Proposition I.1.11], the duaX, X∗ (respectively the predual ofX, X∗, if this exists) is extremely rough. Then, froCorollary 2.9, we deduce the following

Corollary 2.10. BL(X,Y ), the space of bilinear forms onX×Y , is extremely rough, wheneverY be an infinite-dimensionalL1 predual.

Now, we characterize the existence of nonempty weak open subset relative to tball, with diameter less than 2, in a new class of operator spaces.

Theorem 2.11. Let (Ω,Σ,µ) be a positive measure space and letX be a Banach spaceThen the spaceL(L1(µ),X) has a nonempty weak open subset relative to the unitwith diameter less than2 if and only ifL1(µ) is finite-dimensional andX has a nonemptyweak open subset relative to the unit ball with diameter less than2.

J. Becerra Guerrero, G. López Pérez / J. Math. Anal. Appl. 315 (2006) 544–554 551

e

-ying

,

subset

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g

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etern

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l with

Proof. Assume thatL1(µ) is infinite-dimensional. ConsiderAn a sequence of pairwisdisjoint measurable sets with 0< µ(An) < +∞. For eachn we definePn : L1(µ) →L1(µ) given byPn(f ) = f χAn for everyf ∈ L1(µ), whereχAn is the characteristic function of An. ThenPn is a L-projection, that is, a linear and continuous projection satisf‖f ‖ = ‖Pn(f )‖ + ‖f − Pn(f )‖ for every f ∈ L1(µ). Then, following [10], the mapQn :L(L1(µ),X) → L(L1(µ),X) given byQn(T ) = T Pn is a M-projection, that isa linear and continuous projection satisfying‖T ‖ = Max‖Qn(T )‖,‖T − Qn(T )‖ foreveryT ∈ L(L1(µ),X). Now, callingµn = µ|An , we have that

L(L1(µ),X

) =⊕∞

L(L1(µn),X

) ⊕∞ Y,

whereY is a closed subspace ofL(L1(µ),X).PutXn = L(L1(µn),X) andZ = ⊕

∞ Xn. ThenL(L1(µ),X) = Z ⊕∞ Y . In order toprove that every nonempty weak open subset relative to the unit ball ofL(L1(µ),X) hasdiameter 2, it is enough, by Lemma 2.2, to see that every nonempty weak openrelative to the unit ball ofZ has diameter 2. For this, fixz ∈ Z with ‖z‖ = 1. For eachi ∈ N, we definewi(n) = zi(n) = z(n) if i = n and−wi(n) = zi(n) = x0 if i = n, wherexn is a fixed norm one element ofXn for everyn. It is clear thatzi, wi are sequences onorm one elements ofZ weakly converging toz with ‖zi − wi‖ = 2 for everyi so, everynonempty weak open subset relative to the unit ball ofZ has diameter 2.

In the caseL1(µ) is finite-dimensional thenL(L1(µ),X) = n∞(X), that is, the∞-sumof n copies ofX, and it is enough apply Lemma 2.2.

The same above argument work for closed subspaces ofL(L1(µ),X) which are closedunder composition by operators inL(L1(µ)). As a consequence we obtain the followin

Corollary 2.12. Let (Ω,Σ,µ) be a positive measure space and letX be a Banach spaceThen the spaceK(L1(µ),X) of compact operators(respectivelyF(L1(µ),X) of weaklycompact operators) has a nonempty weak open subset relative to the unit ball with diamless than2 if and only if L1(µ) is finite-dimensional andX has a nonempty weak opesubset relative to the unit ball with diameter less than2.

Now, we pass to study the same topic for the spacesL1(µ,X), of Bochner integrableX-valued functions, andL∞(µ,X), of essentially bounded measurableX-valued func-tions, whereX is a Banach space andµ is a positive measure.

Theorem 2.13. Let (Ω,Σ,µ) a positive finite measure space.

(i) The spaceL∞(µ,X) has some nonempty weak open subset relative to the uniwith diameter less than2 if and only ifL∞(µ) is finite-dimensional andX has it too.

(ii) The spaceL1(µ,X) has some nonempty weak open subset relative to the unit baldiameter less than2 if and only ifµ contains atoms andX has it too.

Proof. We can assume, without loss of generality, thatµ(Ω) = 1.

552 J. Becerra Guerrero, G. López Pérez / J. Math. Anal. Appl. 315 (2006) 544–554

to

of

ye unite,of

at

(i) Assume thatµ is atomless. LetW be a nonempty weak open subset relativethe unit ball of L∞(µ,X) and pick f0 ∈ SL∞(µ,X) ∩ W , sinceL∞(µ,X) is infinite-dimensional. Fixδ > 0 and considerW0 = t ∈ Ω: ‖f0(t)‖ > 1− δ. Then, there isWna decreasing sequence of measurable subsets ofW0, with µ(Wn) > 0 and limn µ(Wn) = 0,sinceW0 cannot be an atom ofΩ andµ(W0) > 0. Now −χWn + χΩ\Wn is a sequenceof elements in the unit ball ofL∞(µ) weakly converging to1, the constant function inL∞(µ). Then φn = (−χWn + χΩ\Wn)f0 is a sequence of elements in the unit ballL∞(µ,X) weakly converging tof0 satisfying that

‖f0 − φn‖∞ = 2‖χWnf0‖∞ > 2(1− δ).

Thus diam(W) > 2(1 − δ), sinceφn ∈ W for enough largen. As δ > 0 is arbitrary, weobtain thatW has diameter 2.

In the caseµ has some atom, it is well known that we have, for some setI , the de-compositionL∞(µ,X) = L∞(ν,X) ⊕∞ ∞(I,X), whereν is a measure atomless. BLemma 2.2, the only possibility to obtain nonempty weak open subsets relative to thball of L∞(µ,X) with diameter less than 2 is thatµ be purely atomic and, in this casL∞(µ,X) is not but an∞-sum of copies ofX. The same argument used in the proofTheorem 2.11 shows, that if a∞-sum of copies of a Banach spaceX has a nonemptyweak open subset relative to the unit ball with diameter less than 2, then the∞-sum mustbe finite. The Lemma 2.2 concludes the proof.

(ii) Assume thatµ is atomless. For everyA ∈ Σ with µ(A) > 0, we call rAn the

sequence of Rademacher functions supported onA (see [1, 11.55, 11.56]). We recall th

rAn

µ(A) is a sequence of functions in the sphere ofL1(µ) with support contained intoA

which converges weakly to zero inL1(µ). From the construction of the sequencerAn , it

is clear thatχA+rAn

µ(A) andχA−rA

n

µ(A) are sequences of functions in the sphere ofL1(µ), where

χA is the characteristic function ofA. We putφAn = χA+rA

n

µ(A)andψA

n = χA−rAn

µ(A)for everyn.

ThenφAn andψA

n are sequences in the unit sphere ofL1(µ) weakly converging toχA

µ(A).

Furthermore, one has, for everyn∥∥φAn − ψA

n

∥∥ = 2

∥∥∥∥ rAn

µ(A)

∥∥∥∥ = 2.

Then we have proved the following statement: for everyA ∈ Σ , µ(A) > 0, there areφAn ,

ψAn sequences in the unit sphere ofL1(µ) supported onA, weakly converging toχA

µ(A),

and verifying‖φAn − ψA

n ‖ = 2 for everyn.Pick W a nonempty weak open subset relative to the unit ball ofL1(µ,X). By the

infinite-dimensionality ofL1(µ,X) and the density of simple functions inL1(µ,X), thereis a simple functionϕ ∈ W ∩ SL1(µ,X). We can writeϕ in the way

ϕ =p∑

i=1

χAi

µ(Ai)xi,

wherep is a natural number,Ai1ip are mutually disjoint measurable subsets ofΩ

with positive measure, andxi1ip are vectors inX satisfying∑p ‖xi‖ = 1.

i=1

J. Becerra Guerrero, G. López Pérez / J. Math. Anal. Appl. 315 (2006) 544–554 553

re

all of

if and

exces-al poseodymweaka char-

er.

n

urveys

For every 1 i p let φin = φ

Ain and ψi

n = ψAin be sequences in the unit sphe

of L1(µ) supported onAi , weakly converging toχAi

µ(Ai)and satisfying‖φi

n − ψin‖ = 2, for

everyn.We dofn = ∑p

i=1 φinxi andgn = ∑p

i=1 ψinxi , for everyn. Thenfn andgn are se-

quences in the unit sphere ofL1(µ,X) weakly converging toϕ. Hencefn, gn ∈ W forenough largen and for thisn, we deduce that

diam(W) ‖fn − gn‖ =∫Ω

∥∥∥∥∥p∑

i=1

(φi

n − ψin

)xi

∥∥∥∥∥dµ =p∑

i=1

‖xi‖∫Ai

∣∣φin − ψi

n

∣∣dµ

=p∑

i=1

‖xi‖∥∥φi

n − ψin

∥∥ = 2.

So, we conclude that every nonempty weakly open subset relative to the unit bL1(µ,X) has diameter 2, whenµ is atomless.

It is well known thatL1(µ,X) = L1(ν,X) ⊕1 1(I,X), whereν is atomless andI isa convenient set. By Lemma 2.1 and the preceding paragraph, we conclude thatL1(µ,X)

has some nonempty weak open set relative to the unit ball with diameter less than 2only if I is nonempty andX has it too.

Now, one obtains easily the following consequence.

Corollary 2.14. Let (Ω,Σ,µ) a positive measure space and letX be a Banach space.

(i) The unit ball ofL∞(µ,X) has some denting(respectively weak-norm continuity) pointif and only ifL∞(µ) is finite-dimensional andBX has it too.

(ii) The unit ball ofL1(µ,X) has some denting(respectively weak-norm continuity) pointif and only ifµ contains atoms andBX has it too.

We always have considered in this note, spaces with its natural norm. It would besive to think that the above results hold for equivalent norms. However it seems naturthe following question: is possible renorming every Banach space failing Radon–Nik(respectively point of continuity) property so that every slice (respectively nonemptyopen subset) of the new unit ball has diameter 2? In the affirmative case, it would beacterization of Radon–Nikodym property, by improving the known ones up to now.

References

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