Internat. J. Math. & Math. Sci.
57
(1987)57-61
Vol. I0 No.
ON COMPLEX
LI-PREDUAL SPACES
MRINAL KANTI OAS
Department of Mathematics
University of Nairobi
P.O. Box 30197, Nairobi, Kenya.
(Received November 12, 1985 and in revised form April 30, 1986)
ABSTRACT
This
paper contains some characterisations of complex L-balls,
including interpolation theorems which are analogs of Edward’s
separation theorem for simplices.
KEY WORDS AND PHRASES:
Boundary measures, Choquet ordering, Haar measure,
Lindenstrauss space, Simplex.
1980 AMS SUBJECT CLASSIFICATION CODE. 46 B 99.
1.
INTRODUCTION.
The purpoSe of this paper is to furnish some characterisations of complex L
predual spaces which are being known as Lindenstrauss spaces after
Eli.
1-
The dual
unit balls of such spaces now-a-days called L-balls have been characterised by
many authors includir Lazar & Lindenstrauss
when the spaces are real.
I-2],
Lazar
I-3], I-4],
Lau
I-5]
and others
But their complex versions far from being trivial
follow-ups seem to be much complicated and in reality sometimes need ingenuity to
be formulated even.
This paper contains some complex versions of
E5]
embodied in Theorem 3.
2.
NOTATIONS AND PRELIMINARIES
Lau’s results
For a compact convex subset K of a locally convex Hansdorff space E,
M(K) for the Banach space (with total
stands for the set of its extreme points;
variation as norm)
of complex regular Borel measures on K;
of members of M(K) with norm < i;
eK
MI(K)
for the set
C(K), A(K), P(K) for the space of all real-
valued continuous functions, continuous affine functions, continuous convex functions
on K respectively.
For bounded real-valued functions f on K, the upper envelope is denoted by
f and the lower envelope by f.
II
is said to be a boundary measure if
A measure
is maximal in the ordering of Choquet;
(-f)=O
for all fC(K)
;p.1293.
in fact
is a boundary measure iff
We shall also write
is convex and
If V is a complex Banach space, the dual unit ball K=(V*)
1
We
define the map hom f as (hom f)(x)
compact in the w*-topology.
;F f(x)
d for semi-continuous function f on K,
where d is the unit Haar measure on
(horn f)
(x)
8(horn f) (x) for 8
g
F
F.
Clearly horn f is
F
homogeneous, i.e.
One can easily show that horn restricted to
C (K) are norm-decreasing projections of C(K) onto the space of
F
homogeneous
M.K. DAS
58
continuous functions on K.
The adjoint projection horn defined as horn
=
o hom is
also a norm decreasing w*-continuous projection of M(K) onto a linear subspace
M
horn
(K) of M(K).
where
We can write
f(xei@)d@,
cos@
Slf(x)
(horn f)(x)
7/2
If we write
cos@
(Sf) (x)
(which multiplied by
2(Slf)
(x)
+ i Tf(x)
S f(x)
1
sin@f(xe
Tf(x)
I
)d@
f(xeiO)d@
-/2
n gives what Roy
I]
has defined as f ), then
+
odd Sf(x)
(2.1)
Throughout the paper we shall write A (K) for the set of all continuous
O
F
homogeneous affine functions on K=(V*)
3.
Main Results
i"
V, the following results are recently proved.
For real Banach space
If K is the dual unit ball of a real Banach space V, then the
Theorem i.
following are equivalent:
(i)
V is an
(ii)
If
i, 2
odd
Ul
Ll-predual
space.
are boundary measures on K having the same barycentre, then
odd
U2"
P(K),
(iii) For f
odd
is affine.
(O)
P(K),
x6K}
1/2 sup{f(x) + f(-x):
sup{even
xEK}.
f(x)
(iv)
For any f
(v)
For any l.s.c, concave function f on K such that even f>O, there exists
a continuous affine symmetric function a on K such that f>a.
(vi)
If f, -h are l.s.c, concave functions on K such that h<f and
h(x
xK} < inf{even
f(x)
xeK},
sup{even
then these exists a continuous
affine symmetric function a on K such that h < a < f
The equivalence of (i)
(vi)
(iv)
I-4_
is due to Lazar
while that of
(i),
(v),
_5_
is proved by Lau
Many interesting developments are noted when efforts are made to obtain complex
analogs of these results (many others not stated here) of real Lindenstrauss spaces.
l-8q
A brilliant step towards this have been made by Effros
in complex space.
is to be replaced by horn
Olsen
hypothesis even f > O in (v) is to be replaced by
xSK.
odd
1-93
7.f(kX)>
has shown that the
O for
has tried to give complex analog of
Subsequently Roy
His formulation is rather partial.
by odd(Sf)
who has shown that odd
E93
But
k
F
with
Zk
(iv), replacing
contains some
interesting examples.
Below we give a characterisation of
kind of complex analog of
Theorem 2.
Lau’s result and
complex
Ll-predual
is due to Olsen
space V which is a
E93.
If K is the dual unit ball of a complex Banach space V, then the
following are equivalent:
(i)
V is an
(ii)
For every l.s.c, concave function f on K such that
Ll-predual
all xSK and
k
F,
space;
k
1,2,...,n
with
Z
k=O,
Zf(k x) _> O
there is an a
such that re a(x) < f (x) for all xeK.
We give in Theorem 3, some complex analogs of Lau’s result.
with, we furnish a Lemma below:
for
A (K)
o
However to start
O,
ON COMPLEX L PREDUAL SPACES
1
be a non-zero positive measure on a compact convex subset K
If
Lemma i.
59
of a locally convex Hausdorf space E, then for all u.s.c, convex function f on K
f(x) < (K)
-I
r()
where
x.
rlO; 1.2.2. the stated inequality holds
Mokobodzki(l-lO; 1.5.1) that for every u.s.c, convex
for
By a well-known result
Proof:
f g P(K).
]f(y)d
NOW applying
function f, there is a dscending net
{f
g
f
P(K)}
which converges to f, we
get the desired result.
Our main resuit is
If K is the dual unit ball of a complex Banach space V, then the
Theorem 3.
following are equivalent:
space;
(i)
V is an
(ii)
If f is a l.s.c, concave function on K with even (Sf)(x) > O for all
Ll-predual
K, then there exists an a E Ao (K) such that re a < f on K;
If f, -h are l.s.c, concave functions on K such that h < f and even
x
(iii)
S h (x) < O < even S f(y) for all x,y E K; then there exists an a
E A (K)
O
such that h < re a < f on K.
If f, -h are 1.s.c. concave functions on K such that h < f and
n
g F,
=O}
7
x)
x E K, nE N, O <
sup
k=l
(iv)
k’ 7"k=l’ k
kh(k
7’kk
n
< O < inf{ 7.
k=l
kf(kX):
If g is an u.s.c,
N, O <
x g K, n e
7’k=l’ k gF’ 7"kk=O};
k’
convex function on K, then g(O) <
ak;
7ak=l, keF,
7.a
k
sup{F.g(kX)
O}.
k
(ii).
(i)
Proof
N, O <
K n
e A (K) such that h < re a < f on K;
then there is an a
(v)
x
We shall, in fact, show that (ii) is implied by Theorem 2 (ii).
be l.s.c, concave on K such that even Sf(x)>O.
cos8 f(xe
F(x)
i@)
So let
f
We define
d8
-12
be such that
f
(x)
2 7, even
O.
Tk
Thus
Sf(kX)
Let kE F for k
Clearly F is l.soC, concave.
2 Sf(x).
Then F(x)
O.
7.F(k x)
27.Slf(k x)
7.Sf(kx)
2
which is > O by hypothesis.
there is a function b
We consider the measure
1,2,...,n
+ even Sf(x) and that
Slf(x)
Now note that Sf(x)
+ 27. even
7, Horn
Sf(kx)
Consequently by Theorem 2 (ii),,
A (K) such that re b < F on K.
o
2
I
/2
cos8 E
(xeiS)d8
-/2
where e(y)
is the Dirac measure at y.
By
I-6:
p. ii
On. applying Lemma i, we have 2F(x) < (K) f(x) i.e.
b
we have re a _< f on K.
2
5_
x.
r ()
F(x)
2
<
f(x).
Also (k)
Putting a
4
M.K. DAS
60
(ii)
Let f, -h be l.s.c, concave functions on K such that h < f
(iii).
<O < even Sf(y) for all x,y,
and even S h(x)
E K.
h(x) + h(-x) < O < f(y) + f(-y) for all x,y g K.
We first show that
Let
us establish the last inequality, since the first one can be done similarly.
To
do so we take the measure
2 /2
e(xeiO)d@
cos@
.
-712
and find as before by
6,
1153
p.
that r()
Now apply Lamina 1 to get f(y) + f(-y)
F (x)
Now we define
f (x) A (-h)
Then F is l.s.c, concave.
> O from even
Sf(y) > O.
(-x)
Moreover by the hypothesis and the inequalities just
F(x) + F(-x) > O
proved, we have
4
(K)
x,
for all x g K so that
(K) such that re a < F.
By (ii)
o
function with the desired property.
then we have an a g A
(iii)
,’/2
12 -n12
[h (xe i@
cos@
+ h (-xe
Similarly even (Sf)(y) > 0 for all y
So by (iii) there is an a g
(iv)
(v).
n
x):
go
(g + )
since
x
Since g
ng N,
g K.
such that h
n
K;
_< re
a
_<
f on K.
N; O <
k;
k
i,
F,
k
O}’
7k k
(O) +
F
g(-x).
where (g)(x)
-Og
>
Clearly F is l.s.c, concave.
inf
kF(kX);
Now since g
2g
-
_<
xgK, ng, d> O
Thus g(O)
Note:
o
_<
a, we have g(O)
< (-g) (O)
_<
a(O)
7.kF(kX)
k
k
n
Thus by (iv), there is an hgA
for g
< O
.
(O)
k=l
(v)
(Sh) (x)
+ Og < 2g o by the definition of go’ it follows that g < F. Moreover for
g F,
I,
O, xK, we have
O,
-7.g(-k x)
so that
a(O)
i@)]d@
even
there will be no loss of generality in the assumption
O;
for positive real number
We define
Ao(K)
Then clearly
Suppose that g is an u.s.c, convex function and let
suP{kl kg(k
We assume that
This a is, in fact, the
Let f, -h be two 1.s.c. concave.functions on K, which satisfy
(iv).
the conditions given in the hypothesis of (iv).
go
SF(x) > O.
even
7k=l, k
7k
(K) such that g < re h < F
a(O).
Again
g
O}
-go
We put re h
_> a-2g
O.
A(K)
a
A(K), so that
-g(O).
_<
-g(O) + 2g
(i) is the same as
_7
.
p.
so that g(O)
g(O)<_sup{kg(kX)
--< go and
the result follows
iO3
Our result in (v) is sharper than
P(K),
F,
xgK, nE N,
Roy’s result
O<_, 7.=i,
;
F,
Thin
7
3.3 (iii)
O}.
7
that
k
In fact this follows from (v) immediately, since the reverse
inequality is evident
from the concave character of
LI-PREDUAL
ON COMPLEX
SPACES
61
REMARKS"
In analog with
Lindenstrauss
_2]_
Lazar’s
selection theorem for Choquet simplex, lazar &
formulated a selection theorem for real L-balls which was
]
followed by a complex version by Olsen
analogue of Lau’s result
I]
Our results which are chiefly complex
seem to resemble Edward’s interpolation theorem
s m ces.
REFERENCE
i.
LINDENSTRAUSS, J.
Extension of Compact Operators Memoirs Amer. Math.
2.
LAZAR, A.
LINDENSTRAUSS, J.
Banach spaces whose Duals are Ll-spaces and their
186-193.
Representing Matrices I, Acta Math 126 (1971)
3.
LAZAR, Ao
Space of Affine Continuous Function on Simplex,
Trans. Amer. Math. SOCo 134 (1968), 503-525.
4.
LAZAR, A.
The Unit Ball in Conjugate
Jo 39 (1972), 1-8
5o
LAU, Ka-Sing
The Dual Ball of Lindenstrauss Space, Math. Scand.
Soc. 48 (1964)
&
33, (1973)
Ll-spaces, Duke
Math,
323-337.
(1976).
6.
CHOQUET, G.
Lectures on Analysis, vol. II W.A. Benjamin, INC,
7
ROY, A.K.
Convex Functions on the Dual Ball of a Complex.
Lindenstrauss space, J. London. Math. Soc. (2) (1979)
8
EFFROS, E.G.
On a Class of Complex Banach spaces, Illinois J. Math.
18 (1974)
48-59
9.
OLSEN, GH.
Edward’s separation Theorem for complex Lindenstrauss
space with Application to selection and Embedding
Theorem, Math. Scand. 38 (1976), 97-105.
529-540.
i0. ALFSEN, E.M.
Convex Sets and Boundary
Springer Ver lag.
Compact
I,nte@rals.
(1971);