**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] **

**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) **

**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