C(K), almost CL-space. CL-space On CL-spaces and almost CL-spaces

On CL-spaces and almost CL-spaces

Miguel M a r t i n a n d Rafael Pay~i(1)

Abstract. We find some necessary conditions for a real Banach space to be an almost CL- space. YVe also discuss the stability of CL-spaces and almost CL-spaces by co- and/l-sums. Finally, we address the question if a space of vector-valued continuous functions can be a CL-space or an almost CL-space.

1. I n t r o d u c t i o n

A real or c o m p l e x B a n a c h space is said to be a

CL-space

if its unit ball is the absolutely convex hull of every m a x i m a l convex subset of the unit sphere. If the unit ball is t h e closed absolutely convex hull of every m a x i m a l convex subset of the unit sphere, we say t h a t the space is an

almost CL-space.

T h e c o n c e p t of CL-space was i n t r o d u c e d by R. Fullerton in 1960 [6]. L a t e r on, A. L i m a [12] i n t r o d u c e d t h e almost CL-spaces, t h o u g h the p r o p e r t y specifying t h e m h a d been previously used by J. L i n d e n s t r a u s s [14]. B o t h definitions a p p e a r e d only for real spaces, b u t t h e y e x t e n d easily to the c o m p l e x case. For general i n f o r m a t i o n on CL-spaces a n d a h n o s t CL-spaces, including t h e c o n n e c t i o n with intersection properties of balls, we refer to [8], [9], [11], [13] and the already cited [6], [12]

a n d [14]. More recent results can be found in [19].

E x a m p l e s of real CL-spaces are L I ( g ) for an a r b i t r a r y measure p, a n d its isometric preduals, in p a r t i c u l a r

C(K),

where K is a c o m p a c t H a u s d o r f f space (see [11, C h a p t e r 3]). We do n o t know of a n y e x a m p l e of a real a h n o s t C L - s p a c e which is not a CL-space.

For c o m p l e x B a n a c h spaces, t h e s i t u a t i o n is different. CL-spaces a n d a h n o s t CL-spaces have n o t received m u c h a t t e n t i o n in t h e complex case. T h e only result we are able to find in the literature is the c h a r a c t e r i z a t i o n of finite-dimensional (real or complex) CL-spaees given by A. L i m a in [12, Corollary 3.7].

(1) Research partially supported by Spanish MCYT project no. BFM2000-1467.

108 Miguel Martin and Rafael Payti

In Section 2 we fix our notation and give some examples, mainly in the complex case. We show that the complex spaces

C(K)

are always CL-spaces, whereas com- plex L I ( p ) are just almost CL-spaces. Actually,

ll

and LI[0, 1] are not CL-spaces.

Section 3 contains some isomorphic results. \Xre show that the dual of an infinite-dimensional real almost CL-space always contains (an isomorphic copy of) 1~. If, in addition, the dual is separable, then the space contains co.

In Section 4 we discuss the stability of the classes of CL-spaces and ahnost CL-spaces by Co- and /1-sums. Our results are as expected, the sum is in one of the classes if and only if all the smnmands are. with one remarkable exception: an infinite/i-sum of nonzero complex Banach spaces is never a CL-space.

Finally, in Section 5 we deal with the space

C(K, X)

of continuous fimctions from a compact Hausdorff space K into a Banach space X. We show that

C(K, X)

is an almost CL-space if and only if X is. The analog for CL-spaces remains open.

2. N o t a t i o n and e x a m p l e s

T h r o u g h o u t the paper, X denotes a real or complex Banach space,

B x

is its closed unit ball, S x its unit sphere, and X* the dual space. We will denote by T the unit sphere of the scalar field. Thus, T = { - 1 . 1 } when dealing with real spaces, while T = { A E C : I A I = I } in the complex case. Given a subset

A c X , we

write e x A for the set of extreme points in A and c o A for the convex hull of A. Note that co T A is the absolutely convex hull of A. The closed convex hull of A is denoted by U6A. Finally, for a set

B C X * ,

we denote by /~*~' and U6 .... B the weak*-closure and the weak*-closed convex hull of B.

It is worth pointing out some basic facts on the definitions of CL-spaces and almost CL-spaees. By using the H a h n - B a n a c h and Krein Mihnan theorems, one can easily prove that every maximal convex subset F of S x has the form

F = {z E B x : z * ( z ) : 1 }

for some z * C e x B x . . We denote by m e x B x - the set of those

x * r

with the property that the set

{ x E B x : z * ( z ) = I }

is a maximal convex subset of

Sx.

It is easy to see that m e x B x , is a b o u n d a r y for X. that is. for every

z E X ,

there is

x * C m e x B x ,

such that

x*(x)=ll<l.

It follows that

BX* = Ud u.*

mex Bx* 9

W i t h the above facts in mind, X is a CL-space (resp. an almost CL-space) if and only if, for every

x * C m e x B x . , B x

is the convex hull (resp. closed convex hull) of

the set

{ z E B x : l x * ( x ) l = l } .

Let us also mention t h a t m a x i m a l convex subsets of

S x

are nothing but m a x i m a l faces of

B x .

This accounts for the relation of CL- spaces and ahnost CL-spaces to the facial structure of balls. More information on this topic m a y be found in [1] and [11].

We shall now c o m m e n t on some examples of CL-spaces and almost CL-spaces.

Real CL-spaces are related to an intersection p r o p e r t y of balls, namely the 3.2-

intersection property.

A Banach space is said to have this p r o p e r t y if every set of three mutually intersecting balls has n o n e m p t y intersection. T h e 3.2-intersec- tion p r o p e r t y was first investigated by O. Hanner [8] and systematically studied by J. Lindenstrauss [14] and ft. L i m a [11]. More references on the 3.2-intersection property are [9], [12] and [13]. In 1977. A. Lima showed t h a t every real Banach space with the 3.2-intersection p r o p e r t y is a CL-space [11, Corollary 3.6], but the converse is false even in the finite-dimensional case (see [8, R e m a r k 3.6] or [19, C h a p t e r 3]).

T h e real space L1 (Ft) and its isometric preduals satis~" the 3.2-intersection property.

Therefore, they are CL-spaces.

To end up with the real examples, let us pose an open question. We do not know if there exists a real almost CL-space which is not a CL-space.

The situation in the complex case is quite different. Not much attention has been paid to complex CL-spaces and almost CL-spaces in the literature, so we provide the basic examples. As a m a t t e r of fact. there are complex almost CL- spaces which are not CL-spaces.

P r o p o s i t i o n 1. (i)

For every compact Hausdorff space K. the complex space C(K) is a CL-space.

(ii)

For every finite measure it, the complex space L1 (~) is an almost CL-space.

(iii)

The complez spaces l~ and LI[O,

1]

are not CL-spaces.

We need a l e m m a on the field C whose proof is straightforward. We write D----{AEC:IAI<I}.

L e m m a 2.

For every

z 0 E D ,

there exist two continuous functions

c2, g): D - + D

satisfying

I ~ ( z 0 ) l = l ~ ( z 0 ) l = l

and z= 89 for every zED.

Proof of Proposition

1. (i) By the well-known characterization of extreme points in

BC(K). ,

it suffices to show. for each

toEK,

that

BC(K)

is the convex hull of the set

{ f E B c ( K ) : l f ( t o ) l = l } .

Indeed, given

toEK

and

f E B c ( K ) ,

we take

zo=f(to)

and find functions p and ~;, as in the above lemma. Then

f = 8 9

where

f t = F o f

and

f2=v)of

satisfy

Ifl(to)l=lf2(to)l=l.

(ii) Let F be a m a x i m a l convex subset of

SLy(,).

Up to an isometric isomor- phism, we can suppose t h a t

F = { f E Ll(/.t):

Ilfll

= 1 and f > 0 a.e.}.

110 Miguel M a r t i n a n d Rafael Payti

B u t the absolutely convex hull of F contains a dense subset of the unit ball. n a m e l y t h e set of simple functions, so

BL~(~,)=~6

T F as desired.

(iii) It is clear t h a t exBr~[0.1]. =mexBLl[0.1]~. so tile set F = { f E LI[0, 1]: Ilfll = 1 a n d f _ > 0 a.e.}

is a m a x i m a l convex subset of

SL~[Odl.

T h e n . it suffices to find a function f E

BLI[O,1]

which c a n n o t be o b t a i n e d as an absolutely convex c o m b i n a t i o n of elements in F . Indeed, let

f ( t ) = e 2~it

for every t e l 0 . 1 ] , a n d suppose t h a t we could write f

= ~ j = l a j A j f j

n a.e. in [0,1], where

aj>O.

I)~jl=l.

f i E F

for j = l . 2 .... ,n, and

~ j = l aj

n = 1 . Then, from

1 = f(t)e -2~it d t = oj aje-2~'~fj (t) dt <_ aj fj(t) dt = aj = 1,

= j = l j = l

we would get

A~e-2~itf~(t)=fl(t)

for almost every t E [0, 1], which is clearly impos- sible. T h e case of 11 can be t r e a t e d similarly a n d it will b e c o m e a p a r t i c u l a r case of Corollary 10. []

3. S o m e i s o m o r p h i c r e s u l t s

O u r aim here is to o b t a i n some isomorphic properties of real almost CL-spaces.

We s t a r t with an easy but useful lemma.

L e m m a 3.

Let X be an almost CL-space. Then Ix**(x*)l=l for every x**E

e x B x * *

and every x*

E m e x B x * .

Proof.

If x* E m e x B x . , we have

B x = U 6 { x E B x : l x * ( x ) l = l }

a n d Goldstine's t h e o r e m gives

B x . . = ~ " " {x e B x : Ix*(x)l = 1}.

Now, by t h e 'reversed' Krein M i h n a n t h e o r e m we get e x B x . , c_ {.r c B x : I~'*(~')1 = 1} "

which clearly implies I x * * ( x * ) l = l for every

x**EexBx--. []

Remark

4. I n t h e real case, t h e converse of the above l e m m a holds. This follows from [12, T h e o r e m s 3.1 and

3.4].

A useful sufficient condition for a real B a n a c h space X to contain (a subspace isomorphic to) co or ll was found in [16], n a m e l y tile existence of an infinite set

A C S x

where all e x t r e m e points in

B x .

can only take the values +1. This fact combined with L e m m a 3 yields t h e main result in this section.

T h e o r e m 5. Let X be an infinite-dimensional real almost CL-space. Then X * contains 11. If in addition X* is separable, then X contains co.

Proof. Being a b o u n d a r y for the infinite-dimensional space X. the set mex Bx*

must be infinite. Then, we m a y a p p l y L e m m a 3 and [16. Proposition 2] to get t h a t X* contains either co or ll. But a dual space contains l~c (and hence 11) as soon as it contains co (see, e.g., [15, Proposition 2.e.8]).

To prove the second p a r t of the proposition, observe that L e m m a 3 implies t h a t II x * - y * II = 2 for distinct x*, y* E m e x B x - . Therefore, if X* is separable, mex Bx*

has to be countable. But a real infinite-dimensional space which admits a countable b o u n d a r y contains Co by a result due to V. Fonf [5, R e m a r k 2]. []

Remark 6. T h e above result improves those given in [16] for real Banach spaces with numerical index 1. A Banach space has numerical index 1 if every bounded linear operator T: X - + X satisfies

IITII =sup{Ix* (Tx)l : x E S x , x* E S~;- and x*(x) = 1}.

For more references and background we refer the reader to [2], [3], [4], [17] and [18].

Almost CL-spaces have numerical index 1 [17, C h a p t e r 4]. and it is proved in [16]

t h a t the dual of an infinite-dimensional real Asplund space with numerical index 1 contains l~.

T h e behaviour of CL-spaces or ahnost CL-spaces under duality, is unclear to us. It was shown by A. L i m a [12, Corollary' 3.6] that a real Banach space X is an almost CL-space provided t h a t X* is a CL-space. We are not aware of any relevant result in the opposite direction. Next we get such a result, under an additional assumption.

P r o p o s i t i o n 7. Let X be a Banach space .not containing 11. If X is an almost CL-space, then X* is an almost CL-space.

Proof. For x * * E m e x B x * * , we need to show that B x - = ~ - 6 H . where H : {x* 9 B x - : Ix**(x*)l = 1}.

We have m e x B x . C_H by L e m m a 3. Moreover, since m e x B x , is a b o u n d a r y for a space X not containing ll, we can use [7, T h e o r e m III.1] to get t h a t B x . = c-6 mex B x . , and we are done. []

4. Sums of CL-spaces and almost CL-spaces

This section is devoted to study, the stability" of the classes of CL-spaces and almost CL-spaces by Co- and /l-sums. Given an arbitrary" family {Xa:AEA} of

112 Miguel Martfn and Rafael Payfi

Banach spaces, we denote by [(~.xc.,

Xx]~ o

(resp. [(~.xea Xx] h) the c0-sum (resp.

/1-sum) of the family. In case A has just two elements, we use the simpler notation

X |

o r X @ I Y .

W i t h respect to c0-sums, we have the following result.

P r o p o s i t i o n 8.

Let

{Xa:AEA}

be a family of Banach spaces and let X =

[(~aEaXa]c0.

Then, X is a CL-space (resp. an almost CL-spaee) if and only if Xx is a CL-space (resp. an almost CL-space) for every

AEA.

Pro@

For each AcA, we write Ix tbr the injection of X~ into X. and P;~ will be the projection of X onto Xx. Under the natural identification X * = [(~a~a X~,]l , P;~ becomes the injection of X{ into X*. Using the well-known fact t h a t

AEA

one can easily check t h a t a convex subset F of S x is m a x i m a l if and only if there exist AEA and a m a x i m a l convex subset Fx of

Sx~

such t h a t

(1)

F x = P x ( F )

a n d F = p ~ I ( F x ) N B x .

Suppose first t h a t every X;~ is an almost CL-space, and fix a m a x i m a l convex subset F of

S x .

Then, we can find A EA and a m a x i m a l convex subset Fx of

Sxx

satisfying (1), so

Bx~

= ~ 6 T F a . Now, given

z E B x

and e > 0 . take y a E c o T F a such t h a t

Ily~-P~(z)ll<c,

and consider

y = z - I x ( P x ( z ) ) + I x ( y a ) .

It is clear t h a t

y E B x

and P~(Y)=Yx, so y E c o T F . Moreover,

Ily-zll

<e, so we have shown t h a t

B x

=~6 T F and X is an almost CL-space. If every Xx is a CL-space, then we can repeat the above argument with c = 0 to get t h a t X is a CL-space.

Conversely, suppose t h a t X is an almost CL-space. Fix AEA and let Fx be a m a x i m a l convex subset of

Sxx.

Then, the set F given by (1) is a m a x i m a l convex subset of

S x ,

so

B x = ~ d T F .

Now, given

xxEBx~

and s > 0 , there exists y E c o T F such that

Ily-I~(x,xOll<s.

T h e n

Px(y) E c o T F x

and I I P x ( y ) - x x l l < ~ .

so B x x = ~ - 6 T F x and Xx is an almost CL-space. The same argument with c = 0 gives t h a t X~ is a CL-space if X is. []

T h e examples in Proposition 1 tell us t h a t the situation f o r / 1 - s u m s cannot be so tidy. Actually, we have the following result.

P r o p o s i t i o n 9.

Let

{X~:AEA}

be a family of nonzero Banach spaces and let

X = [ ( ~ c a Xx]h"

Then the following statements hold:

(i)

The space X is an almost CL-space if and only if every Xx is.

(ii)

In the real case, X is a CL-space if and only if every X~ is.

(iii)

In the complex case, X is a CL-space if and only if every X:~ is and the set A is finite.

Pro@

Once again we write I~ and P~ for the natural injections and projections.

Now, X * = [(~aEa X~]l~, I ; is the natural projection of X* onto X l , and the relationship between extreme points in

B x .

and Bx~ is the following:

x * E e x B x . ~

I~(x*)EexBx;,

f o r a l l A E A .

W i t h this in mind, it is easy to check that maximal convex subsets of

S x

are exactly the sets of the form

(2) F = { x C S x :P:~(x) E IIPA(x)IIFA

for all AEA}, where FA is a maximal convex subset of

Sx~

for all AEA.

(i) Suppose first that X is an almost CL-space and. for fixed pEA, let F . be a maximal convex subset of S x . . For every ACp we choose an arbitrary maximal convex subset Fx of Sxx and consider the maximal convex subset F of S x given by (2). Note that

P.(x) E IIP.(x)IIF.

C c o T F . for all x c F , so

P , ( c o T F ) C_ co T F , . Now, using that B x =g-d T F , we have

B x , = P~ ( B x ) = P~ (~5

T F ) C_ P / ( c o T F ) C ~-5 T F , ,

and we have shown that X~ is an almost CL-space. Note that if X is a CL-space, then

B x , = P~

(co T F ) C_ co

TF~,

and X~ is a CL-spaee.

Conversely, suppose t h a t every X~ is an almost CL-space, and let F be a maximal convex subset of

S x .

Then F has the form given in (2) where F~ is a maximal convex subset of

Sx:~

for each AEA. Since

Bxx

= ~ T F A and

IA(F~,)CF

for every AEA, we have

U I~(Bx~) C U

~ T I ~ ( F ~ ) C~-STF.

AEA AEA

114 Miguel Martfn and Rafael Payfi

Now, just recall that

B x

is the closed convex hull of

UhEAIh(Bxx),

So BXC_

~ 6 T F and X is an almost CL-space. Observe that, if A is finite, then

B x =

co Uhe A I h ( B x x ) and the above argument allows us to obtain

B x = c o T F

from

Bxx

= c o T F h for all A. Therefore, finite

ll-sums

of CL-spaces are CL-spaces.

(ii) One implication has already been proved. For the converse, suppose that each Xh is a real CL-space, let F be a maximal convex subset of

S x

and write F as in (2). Fix

X E S x

and use that

Bxx

= c o ( F a U - F x ) to write

ex(x) = llPh(x)li(thyh--(1--t~)zh). X e A .

where Yh,

zaEFa

and O_<ta_<l for all A. It follows that x = Z = u - ~ , ,

AEA where

y = IlPh(x)llt ,5(Y ,) a n d =

IIPh(x)Jl(1--th)5( h)

AEA AEA

satisfy that

IlYll+llzll = ~ IIPh(x)ll = 1.

hE:\

For each A we have

PhQY)=llPh(.~)llYhellP~,(Y)llfa

and it clearly follows that we may write

y=llyllyo

^with

yoEF.

Similarly,

z=llzllz0

^with

zoCF,

and we get

x = Ilylly0-II llz0 c c o ( f u - F ) .

We have shown that

S x

C c o ( F U - F ) , so

B x

= c o ( F U - F ) and X is a CL-space.

(iii) We have already seen that every X~ is a CL-space if X is. and that the converse holds for finite A. Thus. it only remains to show that X cannot be a CL-space if h is infinite. W i t h o u t loss of generality we just consider the case A = N . Indeed, choose an infinite countable subset A0 of A and write

X = Y - G 1 Z

with Y = IEI~hEAo Xh] h and suitable Z. Were X a CL-space. our above results show that Y would be a CL-space as well. Thus. suppose that X = [(~-EN Xk]l~ is a CL-space to get a contradiction.

For each k E N , we fix

e~.EmexBx;

and build up the set

F = {x E S x : e~.(Pk(x)) = HPt,(x)H

for all k E N}, a maxirnal convex subset of

S x .

Now. let

x = 9~#Ik(ek).

k = l

Og O(2

where ( k)k=l is a sequence of distinct elements in T, eA.ESx, every k E N . Since B x = c o T F , we can write x in the form

= } 2

j = l

where

/~1, ,22,..., 3rn ~ C, ~ 13j[ : 1

j = l

We deduce that, for every k E N ,

SO

1 = ~ =

and x (1). x (2) ... x ('') E F.

t i t rrl

C~ k

2~: = e~(pk(x)) = ~ 35~;(rk(x<J>)) = ~ ~j IIPk(x(J))II,

j = l j = l

j = l k 1 j = l

k = l j ~ l k = l

It follows t h a t

(3) ~ZjllPk(x(5))ll = 13jl IlPk(x(J))ll

for every j E { 1 , 2, ..., m} and every k E N . Since

Tn

o # P~(x) = ~ 3jP~.(x(J))

and e~,,(ek)=l for

IIx(J) II = 1.

LI(p)-ll(F)@I [(~ LI([O, 1]

^{... )]}

aE.4 I~

j = l

for each k c N , there must exist some j E { 1 , 2 ... m} such that 3jPk(x(J))7~O, and (3) gives ak=/3j/i/3jl. This contradicts the choice of the sequence (ak)~=l. []

Let us point out the following consequence.

C o r o l l a r y 10. Let > be a measure. The complex space L I ( # ) is a CL-space if and only if dim L1 (#) < oc.

Proof. By [10, pp. 136], we have

116 Miguel Martin and Rafael Pay&

for some index set F and some set A of cardinal nmnbers m~_>R0. If L1 (/1) is a CL- space, Proposition 9 gives us t h a t F and A are finite and LI([0, 1] ~ ) is a CL-space for every c~EA. Now, since T h e o r e m 14.10 of [10] gives

LI([0, 1]m~ ~ LI([0, 1] m~ ,

-- l l

where rnk=rn~, for every k E N , another application of Proposition 9 tells us t h a t LI([0, 1] " ~ ) is not a CL-space. Therefore, A = 0 , L I ( t 0 ~ l l ( F ) for a suitable finite set F, and d i m L l ( t t ) < o c . T h e converse result is clear. []

5. T h e s p a c e

C(K,X)

Given a compact Hausdorff space K and a Banach space X. we consider the Banaeh space C(K, X ) of all continuous functions from K into X, endowed with its natural s u p r e m u m norm.

P r o p o s i t i o n 11. Let K be a compact Hausdorff space and let X be a Banach space. Then the following statements hold:

(i) The space C(K, X ) is an almost CL-space if and only if X is.

(ii) if C(K, X ) is a CL-space, then X is also a CL-space.

Proof. Let us write Y = C ( K , X ) and recall t h a t the extreme points in Bz* are functionals of the form f~+x*(f(t)), where x * E e x B x , and t E K (see [20, Theo- rem 1.1]). W i t h this in mind, it is easy to check t h a t the m a x i m a l convex subsets of S y are just the sets of the form

(4)

= { f E S y : f ( t ) E F},

where t E K and F is a maximal convex subset of S x .

Suppose first t h a t X is an almost CL-spaee. Fix a m a x i m a l convex subset ~- of Sy and let t and F be as in (4). Since B x = ~ d T F , given f E S y and e > 0 we can find z Eco T F such t h a t

IIx- f(t)ll < -c and build a continuous function ~2: K--+ [0, 1] such that

~ z ( t ) = l and ; ( s ) : 0 , if I I x - f ( s ) l l ~ . If we define g E Sy by

g(s) = 9 ~ ( s ) x + ( 1 - ~ ( s ) ) f ( s ) , s E K.

it is e a s y t o see t h a t g E c o T $ r a n d IIf-gll<_a. T h e r e f o r e . B y = U 6 T . T a n d Y is a n a l m o s t C L - s p a c e .

Now, s u p p o s e t h a t Y is a n a l m o s t C L - s p a c e . F i x a m a x i m a l c o n v e x s u b s e t F of S x , a n d c o n s i d e r t h e m a x i m a l c o n v e x s u b s e t b r of S y given b y (4) for an a r b i t r a r y t E K . F u r t h e r , fix x E B x a n d ~ > 0 , c o n s i d e r f = X X K E B y a n d use t h a t B ~ = K d T 5 c t o find g E c o T 9 c such t h a t

Ilf-gll

<~- T h e n g ( t ) E c o T F a n d

IIg(t)-xH <~,

so

Bx

=

~6 T F a n d X is a n a l m o s t C L - s p a c e .

I n t h e c a s e w h e n Y is a C L - s p a c e , we c a n r e p e a t t h e a b o v e a r g u m e n t w i t h ~:=0 t o p r o v e t h a t X is also a C L - s p a c e . []

To finish t h e p a p e r , let us m e n t i o n t h a t we d o not k n o w w h e t h e r C ( K , X ) is a C L - s p a c e w h e n e v e r X is.

Acknowledgement. T h e a u t h o r s wish to e x p r e s s t h e i r g r a t i t u d e to t h e referee for p o i n t i n g o u t C o r o l l a r y 10.

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Received November 15, 2002 in revised form February 26, 2003

Miguel Martfn

D e p a r t a m e n t o de Anglisis Matemgttico Facultad de Ciencias

Universidad de G r a n a d a ES-18071 G r a n a d a Spain

emaih mmartins,~ugr.es Rafael Payfi

D e p a r t a m e n t o de Anfilisis Matemfitico Facultad de Ciencias

Universidad de G r a n a d a ES-18971 G r a n a d a Spain

email: r p a y a ~ u g r . e s