ON M-IDEALS AND THE ALFSEN -EFFROS STRUCTURE TOPOLOGY

ABSTRACT

The M-ideals of a G-space are characterized, and it is shown that a Banach space V is a G-space if and only if the Alf.sen- Effros structure topology on the extreme points of the dual ball is Hausdorff.

IN"TRODUCTION

Let V be a real Banach space and aev1 the extreme points of the unit ball of V* ^o Alfsen and Effros used in [3] the w*- closed L-summands of V* to define a structure topology on a e V-1 ^o In this topology a ⁿ E

a

V*

.l"' e 1 can never be separated from its nega-

ti ve, hence is it sometimes more convenient to identify p and - p and use the quotient space (aeV.1)a.. The Alfsen-Effros structure topology has two important special cases:

1) If K is a compact convex set in a locally convex Hausdorff space, then there is a facial topology on ^{6 K}

e (see [1] or [2]).

Let V = A(K) (the set of all continuous affine functions on K ) , then it is konwn that (

o

e V

1)

^a is homeomorphic to a ₈K .,

2) If V is a Lindenstrauss space (i.e. V* isometric to an L₁ (f..l )- space), then the structure topology on aeV1 introduced by Effros in [5] coincides with the Alfsen-Effros structure topology (see [3]

p. 168).

A subspace N of V is an L-summand if there is a subspace N' of V such that NnN'

=

[0}, N+N'

=

V, and for each pEN,

q E N'

l\p+q\1

= IIP!I

+

II

^qll

From the symmetry of the definition we see that N' is an L-sum- mand. N' is unique and hence we call it the complementary L-sum- mand of N.

ABSTRACT

The 1"1-ideals of a G-space are characterized, and it i.s shown that a Banach space V is a G-space if and only if the Alf.sen- Effros structure topology on the extreme points of the dual ball is Hausdorff.

INTRODUCTION

Let V be a real Banach space and oeV1 the extreme points of the u_nit ball of V* .. Alfsen and Effros used in [3] the w*- closed L-summands of V* to define a structure topology on o e

V.1 ..

In this topology a p E oeV1 can never be separated from its nega- tive, hence is it sometimes more convenient to identify p and - p and use the quotient space ( o e V-1) ₀ ^o The Alfsen-Effros structure topology has two important special cases:

1) If K is a compact convex set in a locally convex Hausdorff space, then there is a facial topology on

o

K (see [1] or [2]).

e

Let V

=

A(K) (the set of all continuous affine functions on K ) , then it is konwn that (oeV1)₀ is homeomorphic to o₈K.,

2) If V is a Lindenstrauss space (i .. e. V* isometric to an L₁ (f.l )- space), then the structure topology on oeV1 introduced by Effros in [5] coincides with the Alfsen-Effros structure topology (see [3]

p .. 168) ..

A subspace N of V is an L-summand if there is a subspace N' of V such that Nn N'

=

(0}, N+N'

=

V, and for each p EN,

q EN'

!\p+q!J

= II PI!

+

II

^q\1

From the symmetry of the definition we see that N' is an L-sum- mand. N' is unique and hence we call it the complementary L-sum- mand of N.

A closed subspace J of V is an M-ideal if its annihilator J⁰ is an L-summand of V* " In Corollary

5

we give a condition on the M-ideals of V that is sufficient to ensure that V is a predual L1(~)-space. We then use an analog of this condition or rather its equivalent formulation on the w*-closed L-summands of V* , to define a separation axiom for topological spaces. We call it the splitting property and it follows that (Corollary

7)

V* is isometric to an ^L1^(~)-space if (oeV1)₀ has this property. The splitting property is stronger than T₁ and weaker than Hausdorff"

In § 2 we characterize the M-ideals of a G-space (Theorem 9)"

We show that a closed subspace J of V ( ⁼ [f ^E C(X) :f(x₀₎ =

Aaf(ya)}) is an M-ideal if and only if

J

=

[f E V ~ f(x)

=

0 for all x E F} for some closed set F c X.

Theorem 10 is our main result, We there generalize r·esults ([2], Theorem 6.2 and [5], Theorem 6o3) on the facial topology of

o

K

e and the structure topology on (oeV1)₀ ( V a predual L1(~)-space),

We show that a Banach space V is a G-space if and only if

(o e V~) 1 a is Hausdorff, We also show that in a G-space the inter- section of any family of M-ideals is an M-ideal, and we raise the problem iiJhether the G-spaces can be characterized in this way, Finally in Theorem 12 we show that (oeV1)₀ is perfectly normal if V is a separable G-space.

In § 3 we give some examples of Banach spaces where (oeV1)₀ has the splitting property, and examples of f~ilies of M-ideals such that their intersection is not an M-idealo

Part of this paper is from the authors cand,realo thesis pre- pared in the period

1974-76

at the University of Oslo under direc- tion of professor Erik Alfsen" Most of it is based on .Asvald Lima's paper ¹¹Intersection properties of balls and subspaces in

Banach spaces^{11 ,} ['1'1], and many of the results must be regarded as corollaries of his results. The author wants to thank Erik Alfsen, Asvald Lima and in particular Gunnar Olsen for encouragements,

discussions and valuable suggestionso

1o L-SUMMANDS AND L1(~)-SPACES

In [11], Theorem

5.8

has Lima given several characterizations of Lindenstrauss s:pc:.ces., To make the geometrical content clearer we shall here reformulate one of themo First we need some l8mmas about L-summands.

LEMMA 1 ⁰ Let L,N and N be L-summands in a Banach space

v.

Then:

(i) If N()L = [0} then L c N' (ii) If N+L

= v

then N' CL

(iii) If N()L = [0} and N+L ==

v

then L

=

N'

(iv) If NnL

=

^{[0} and N()M} = ^{[0} then N}

n

(L+M) = ^(0}

(v) If NcL then there exists an L-summand M such that N+M ⁼ L and

Nn

M ^{= [0}}

PROOF. The definition of an L-summand and simple verification.

LEMMA 2.. Let N be an L-summand with dim N ~ 2 .. Then N can be written as a direct sum of two L-summands both different from N if and only if there exists an L-summand L such that L c N and [ 0 } ,£ L /:. N •

PRO ^OJ!'. If N

=

L + M a..r1d L, M /:. N , then L c N and [ 0} /:. L

f

No Conversely if L c N and [0} /:. L /:. N , then from Lemma '1 ( v) there is an M such that N

=

L + M , L

n

M

=

[0} and M /:. N ^o

THEOREM 3. Let V be a real Banach space. Then the following statements are equivalent:

(i) V* is isometric to an L1(~)-space,.

(ii) [0,'1]oeV1

= n {

(L

u

L ^{I )} n

v 1 :

^L is an L-summand in V*} and span(p) is an L-summand for all p E oeV1 ..

(iii) If N is any L-summand in V* of dimension > 2 , then N can be written as a direct sum of two L-summands both dif- f erent from N •

PROOFo The equivalence between (i) and (ii) is proved in [1~]

Theorem 5.8.

(ii) ~ (iii) Suppose (ii) is true, and suppose there is an L- summand N in V* of dimension > 2 that cannot be written as a direct sum of two smaller L-summands. Let L be any L-summ~~d

in V* .. We cannot have N n L /= [0} and N

n

^L

.J

N according to Lemma 2, hence N c L or N

n

^L

⁼

[0} " Then from Lemma '1 (i) N c L and N S:,: L^{1 ,} that is N c LUL^{1 •} Thus

Nn V*

'1 ^~

n [

(L

u

L ^{I )} n

v1 :

L is an L-su.mm~'>ld in V*} and from

(ii) Nn V* c [0 '1]o V*

' 1 - ' e'1" Let p E N and

liP!! ₌

'1 ' then u E o V1 _~ e ^o From (ii) span(p) is an L-summand, and [ 0 } .j span ( p ) .j N • But this is impossible by Lemma 2, and we have got a contradiction.

(iii) =? (ii) Suppose (iii) is true. Since we always have

[ 0 ' '1

J

0 e

v 1

^{c n { ( L}

^u

^{L I ) n}

^v 1 :

^L is an L- summand in

v

* } (

l

'1 '1

J

p.., 34) i t is enough to prove the converse inclusion. Suppose then

p %. [0,'1]oeV1· We may assume JJpJJ ~ '1.. Let N(p) be the inter- section of all L-surnm.ands containing p • Then N(p) is an L- summand ( [ 3] Prop ^Q I. 1 ^Q '13). Dim N(p) > '1 since p

.J

0 • If

dim N(p)

=

'1 , then N(p)

=

span(p) and N(p)

n

oeV1

=

oe(N(p) n

v1)

=

{±JJp\J-'1p} , hence

IIPII-

1P E o V* e '1 • But that is impossible since

p

f.

[0,'1]oeV1., Hence dim N(p) .:::_ 2 and there are L-summands N and I1, NnM ⁼ {0}, N+I1

=

N(p) and N,I1

I

N(p). Now p y!N and p f.M since N(p) is the smallest L-summand containing p • Define L

=

N + N(p)' , then L is a.c"''l L-summand and p

f.

L since p E N(p) and p

f.

N • It is also easy to prove (by using Lemma '1) that L ^{1 =} I1 and hence ^p

f.

L' • Together ^p

f.

L U L ¹ and we have p

f.

n [(L

u

L I) n V-1 : L is an L-sum.ma."'ld in V*} •

It remains to prove that span(p) is an L-summand for all p E o e V-1 .. Let N(p) as before be the smallest L-sum.mand con- taining p • Suppose dim N(p) .:::_ 2 .. Then there are L-summands N, M, N + I1

=

N(p) and N, I1

I

N(p) • We always have p E N(p) n o e V-1

= (N+I1) n oeV-1 = (Nn oeV-1) U (Mn oeV-1). Hence ^p EN or ^p E I1.

But that contradicts the choice of N(p) ^o Thus dim N(p)

=

'1 , N(p) = span(p) and span(p) is an L-sum.mand.. The proof is com- plete.

We are now turning to the w*-closed L-summands in V* • DEFINITIONa We say that we can split a w*-closed L-summand N in V* if there exist w*-closed L-summands L and I1 , L, I1

I

N such that N = L + I1.,

REMARIL A simplex K is said to be prime (see [1] p. '164) if for any two closed

sarily F1 ^{= K} Let F, F1 and We say that F

Let

faces F1 'F2 such that K ⁼ conv(F1 U F2 ) , neces- or F2 ^{= K.} This definition can be ext ended.

F2 be closed splitfaces of a compact convex set K.

is prime if F

=

conv(F 1 U F 2 ) implies F 1

=

F or J be the M-ideal in V

=

A(K) defined by (see [3]

F 2 =F.

p. 100) J

=

[a E A (K) : a ( x) = 0 for all x E F} , and let N = J • ⁰ Then we can split the 'II'J*-closed L-sum.mand N if a.TJ.d only if F is not prime.

THEOREM 4., Let V be a real Banach space, then V* is isometric to an L1^(~)-space if we can split every w*-closed L-summand in V * of dimension .:::, 2 ^o

PROOF.. Let N be any L-summand in V* with dim N > 2 and let N* be the intersection of all w*-closed L-summands that contains N (there does exist one since V* is an L-summand) .. N* is a w*-closed L-summa.nd, and dim N* > 2 since N ~ N* • Hence there

exist w*-closed L-summands L and M , L, M -} N* and L + M = N* ..

Since N* is the smallest w*-closed L-summand containing N , we have N

£

L and N

£

11 o Now N n L -J [0} or N n M -} [0} because if Nn L = [0} and Nn M = [0} then from Lemma 1(iv) [0} =

N

n

(L+M) = N n N* = N ^o Without loss of generality we may assume N

n

L -} [0} , hence [0} -} N n L -J N .. By using Lemma 2 we verify

statement (iii) of Theorem

3,

and V* is isometric to an L1^(~)- space ..

If J is an M-ideal in a Banach space V then the annihi- lator J⁰ is a w*-closed L-summand, and if N is a w*-closed L-summand in V* then there exists an M-ideal J in V such that Hence we can find a property for the M-ideals in V that is an analog to the split property for w*-closed L-sum- mands in V*.

DEFINITION. We say that an M-ideal J c V is reducible if there exist M-ideals J1 and J2 , ^{J -} J}1 ^,J2 such that J

=

^J₁

n

J2 ..

An M-ideal is irreducible if it is not reducibleo (This defini- tion is due to Alfsen.)

COROLLARY 5 .. A Banach space V is isometric to a predual L1^(~)­

space if every irreducible M-ideal J -} V is a hyperplane (i .. e ..

codim J

=

1) ..

PROOFo Use Theorem 4, the comments above and the fact that

J~ + J~

=

(J'1

n

_{J 2 )}⁰ for all M-ideals J₁ and J 2 in V (['1'1],

Lemma 6 ., '18 ) •

DEFINITION. We say that a topological space has the splitting property if for every closed set F that contains more than one point, there exist closed sets F'1, F2 and F'1,F2 /: F such that

F 1 U F2 ⁼ F ..

If Y has splitting property then Y is T'1 , because

1YT

can be written as a union of two smaller closed sets if it contains more than y , and that is impossible since

TYT

is the smallest

colsed set containing y ^a It is not difficult to prove that a T₁-space where all convergent nets have at most finitely many limit- points will enjoy this property, and hence every Hausdorff space has the splitting property.. But such a space is generally not Hausdorffo

LEMMA 6o Let V be a real Banach space., Then (oeV~)0 has the splitting property if and only if we can split every w*-closed L-surnmand in V* of dimension > 2 •

PROOF., A set in

a

_e ^V* ₁ is (structure) closed i.f and only if it is of the form N

n ^a

_{e '1} ^V':' where N is a w*-closed L-summand .. Hence the splitting property on _(oeV~)cr is equivalent to the property that to all w*-closed L-summands N of dimension > 2 there exist v'f*-closed L-summands N₁ and N2 such that

('1.'1) and

('1.2) N. _l

n

^a _{e '1} ^{V* /: N}

n

_· ^o _{e '1} ^V* i='1,2

Suppose ('1 .. '1) and ('1o2) are true. N₁ +N₂ is a w*-closed L-sum- mand (see ['1'1] Lem.m.a 6.'18) and from [3] l i Prop .. '1 .. '15 oe(Nn

v;p ⁼

Nn '\V1 = (N₁ n oev;p

u

(N₂

n

oeV1) = (N₁+N_{2 )}

n

oev1 = oe((N₁+N_{2 )} n v;p and hence N

n v:; ⁼

^(N_{1+N2 )}

n v_:;

(The Krein-Milman theorem)" Now N = N₁ + N₂ since N and N₁ + N₂ are sub spaces, and from ( '1.,2) N₁ ,N2

f.

N,. If conversely N ⁼ N₁ + N2 and N1 ,N2 ^~ N, then i t is trivial to prove ('1.'1) and ('1 .. 2) ..

COROLLARY

7.

Let V be a real Banach space.. V* is isometric to an ^L1^(~)-space if the structure space (oeV1)₀ has the ^spli~ting property.

PROOF. Use Lemma 6 and Theorem 4o

2.. G-SPACES

A real Banach space Vis said to be a G-space if there exists a compact Hausdorff space X and a set

S = {(xa,ya.,A.a)} ~ XxXx [-'1,'1] such that V is isometric to the space A= [fEC(X):f(xa)=A.af(ya)for all (xa.,Ya,A.a)ES],

A G-space is isometric to a predual L1(~)-space, this was first proved by Lindenstrauss (['13] Theorem 6.9). Lima has given a new nroof in ['1'1] Theorem 7.10.

In [3] is a subspace N of V* defined to be hereditary if q E N and

liP\\

+

II

q-p!l =

II

qll implies p E N ..

LEMr1A 8. Let X be compact Hausdorff and V ~ C(X) a Banach space and let J cV be a closed subspace such that J⁰ is hereditary ..

Then there exists a closed set F eX such that J

= {

f E V: f (X) = 0 for all x E F} ^o

PROOF. Let ex, x E X be the point measure and define

F

=

{x EX: ex E J0 } , F is closed since J⁰ is w*-closed.. Now

f E J is equivalent to p(f)

=

⁰ for all p E J .. ⁰ Let f E J and X any point in F ' then ^E: X E Jo and hence ⁰ = ^E: (f)= f(x),

X

thus f(x)

=

⁰ for all x E F. Assume conversely that f(x)

=

⁰

for all x E Fo Let p E oe(J⁰

nv:p,

then p E J⁰ n o₈

V1

^since

J⁰ is hereditary ((3] Prop. 1I '1.'15) and hence there exists x E F and A. '

i

^A.

I =

'1 , such that p

=

A. e , and so

X p(f) = A. E: (f)

=

X

A. f(x)

=

0 .. From the Kr.ein-l1ilman theorem every q E J⁰ is the w*-limit of a net {p } where each Pa. a. is a linear combination of points from oe(J⁰ n

v;p ,

thus q(f) = 0 and hence f E J,.

I f F is a subset of X and V ~ C(X) , we define

JF

=

(f E V : f(x)

=

⁰ for all x E F} • Let F

-

be the closure of F ,

then JF

=

JF since all f E V are continuous on X.

THEOREM 9o Let V ~ C(X) be a G-space (i.e. V

=

(fEC(X) :f(x_{0 )} = A.a.f(ya)})o A closed subspace J of V is an 1'1-ideal if and only if J = JF for some closed set F ~ Xo

PROOF., "only ifn follows from Lemma ⁸ since J⁰ is an L-summand and hence hereditary. Let F be any closed subset of X, and let J

=

JF ^o It now suffices to prove that J is a semi M-ideal

since all semi M-ideals in a predual L₁(\.l)-space are M-ideals ..

(A consequence of ('11] Theorem 5o5)o Choose any functions f E J , g E V with

II

^f\\ ~ '1 ,

II

^g\1 ~ '1 .. If we now are able to prove

(2 .. '1) JnB(g+f,'1)nB(g-f,1) 1=

0

then we can use Theorem 6. '15 of ( 1'1] (with E: = 0 ) to conclude that J is a semi M-ideal. Define 111 ,h2 and h by h 1 (x)

=

g(x) + f(x), h 2 (x) ⁼ g(x)- f(x) and

h(x) = max(h1 (x),h2 (x),O)+min(h1 (x),h2 (x),O)-g(x), x EX.,

Now h+g E V ([13]Lemma 6.7) and hence hE V .. Let x E F, then f(x)

=

0 and h(x) = max(g(x),O) +min(g(x),O)- g(x)

=

0, and so h E J ^o Let x E X , then

~g(x) if h1^(x)~h2^(x)~O or O:;,h2^(x)~h1 ^(x) h₁(x) -h(x) = ~f(x) if h₁(x)_:::O_::h2 (x) or ^h2^(x).s_O~h1 ^(x) lh1(x)+f(x)

ifh 2 ^(x)~h 1 ^(x)_::O

^or

^O~h 1 ^(x)~h 2 ^(x)

In the third case we have lh1 (x)+f(x)l ^~ max(lf(x)l ,lg(x)l) and hence x since \\f1\ _:::. 1 and \\g\\ ^{< 1 ..}

Thus

ih₁(x)-h(x)l ~ 1 for all

\\h1-h\\ :;, 1 or

II

^{g+f -h\1 ~ 1 and h E B (} g+ f , 1 ) .. Similarily we prove h E B(g-f,1) and so we have (2.1).

REMARK. Let Y

=

[x E X:e Eo _{x e ,} V~} , a set F c _- Y is said to con- tain all its extreme points if

n

[f-1 (0)

n

Y:f E JF}

=

F.. Such a set has to be relatively closed in Y,. Now it follows from Theorem 9 that a closed J c V is an M-summand if and only if J = JF for some relatively open-closed F ~ Y and both F and CF contain all their extreme points. If 0 ~ oeV1 (w*-closure) and X is connected, then V does not contain any nontrivial M-summandso But if 0 E oeV1

-

or X is not connected, we can have, but we need not have, any nontrival M-summandso

THEOREM 10.. Let V be a real Banach spaceo The following state- ments are related in this way: (i) <m> (ii) and (i) => (iii)

(i) V is a G-space

(ii) (oeV1)0 is Hausdorff

(iii) The intersection of any family of M-ideals is an M-ideal and ker(p) is an 1'1-ideal for all p E

o

e V1 ..

PROOFo (ii) => (i)

splitting property and hence by Corollary

7

V* is isometric to

an L1(~)-space. Since the structure topology on (oeV1)₀ coin- cides with the biface topology we can use [5] Theorem 6o3 to con- clude that V is a G-space.

(i) => (ii) If two Banach spaces are isometric, then the structure-

spaces are homeomorphic, hence it is sufficient to prove it for a G-space V ~ C(X) (i .. eo V

=

{f E C(X) :f(x_{0_)}

=

A.af(ya)}).. Let p1^,p2 ^{E oeV1} be linearly inaependent points, then there exist

such that p. = A.. ex. ,

l l l

loss of generality v-Te may assume

I"--

I= 1 ' i=·1,2.,

l

A.1

=

_{A. 2} ⁼ 1 • Choose

Without w*-contin- uous linear functionals, ioe. f,g E V such that f(p_{1 )}

=

g(p_{1 )}

=

f(p 2 ) ^=- g(p2 )

=

1. Define F1 ⁼ {x EX~ f(x) > 0, g(x) 2::.0 or f(x) ~

o,

g(x) .::_ 0} and F2

=

[x EX: f(x) ~

o,

g(x) _:: 0 or f(x) ~0,

g(x) ~ 0} , and Ni

=

JFi , i ⁰

=

1,2.. N1 and N2 are by Theorem 9

w*-closed L-summands. Now (N₁

n

aev:p

u

_N2

n

oeV1) == oeV1 since F₁ UF2 =X, and P· _l E N i

n

o e

v.:; ,

ⁱ

⁼

^{1 , 2 since x}1 E F ₁ and x 2 E F2 ..

g(x) for

Define all X E

h(x)

x ..

=

max(f(x),g(x),O) +min(f(x),g(x),O)- f(x)- Then h E JF2 and h(x^{1 ) =- 1 ,} hence

p₁

f.

_N2

n

oeV1 .. In a similar way we find p 2

f.

N₁

n

oev;;. Thus (oeV1)₀ is Hausdorff ..

(i) => (iii) It suffices to prove i t for G-spaces

V

=

{f E C(X): f(xa) = "-af(ya)}.. Let {JY} be any family of M- ideals in V , then there exists by Theorem 9 a family {F Y} of closed sets in X such that JY

=

JF for each y. Now

y

n

_{y y} J

= n

_y J F _y

=

J UF _y

=

J UF _y and hence

n

^{J y} is an M-ideal by Theorem 9 ^o

y

Let p E oeV1, then p = A.ex' x EX and ker(p) = [fEV:f(x)=O}

= J {x} ^o Hence ker(p) is an M-ideal by Theorem 9 ..

REMARK 1~ (i) => (ii) was proved by Effros [5] in the separable case, and later generally by Fakhoury [7] and Taylor [16]. The main idea in our proof is from Taylor.

REMARK 2. We do not know whether (iii) ~> (i) is true or not~

This is a more general form of a problem raised by Effros ([6]

Po 115) and solved in the separable case by Gleit [9]. He proved that if V is a separable simplex space then V is an M· -space if and only if the intersection of any family of M-ideals is an M-ideal. Statement (iii) can also be formulated in terms of L-

summands in V* that is

'

(w*-closure) is an L-summand for any family [NY} o.f iv*-closed L-summands (this is similar to Stermers axiom for'compact convex sets, see [1]p. 1~6) and span(p) is an L-summand for all p ^E oeV1 ^o Lima has proved such a result for compact convex sets, [12] Theorem 20o

REMARK 3o Roy proves in [15] Lemma 4 that for a G-space V the family [Uf :f E V} where Uf ~ [p E o e V1 :f(p) f O} 1 is a basis for the structure topology on o e V1 • Now it is not difficult to prove that a Banach space V satisfies statement (iii) if and only if the family [Uf :f E V} is a basis for the structure topology on oeV1·

The following corollary was first proved by Alfsen and Andersen (see [1] TheoremTI,7o8 or [2] Theorem 6o2)o

COROLLARY 11o Let K be a compact convex set in a locally convex Hausdorff spaceo Then the facial topology of o K is Hausdorff if

e and only if K is a Bauer simplexo

PROOF.. Let morphic to

V

=

A(K) , then o K with facial topology is homeo-_e (oeV1)cr with structure topologyo Now A(K) is a

G-space if and only if K is a Bauer simplex, and the corollary follows from Theorem 10a

Effros proved as mentioned above, that (oe v.:pcr is Hausdorff if V is a separable G-spaceo Roy has pointed out ([15]pv 145) that a slight change in his proof gives that (o V*) _{e 1} is in fact

a

a normal spaceo We vJill show that (aeV1)cr is perfectly normal, and our prooi is almost a copy of a part of the proof Gleit made for [8] Prop. 1.6.

THEOREl.'1 12 • (oeV1)cr is perfectly normal if V is a separable G-space.,

PROOF. From the above remarks it suffices to show that each closed set is a G_{0 •} Let N be any w*-closed L-summand in V* • The w*-topology on V* ₁ is metrizable, and Gleit constructs the follow- ing metric that generates the w*-topology,.

p,q E V1

where [an} is dense in the unit ball of A(v.:p • Then he defines f(p)

=

d(p, N

n

^v.:p

and shows that this f is a continuous and convex function and f(O) = 0 ^o Define

=

(p E o _{e V1}* : f ( p ) _-n >

.1}

then each en is structurally compact in oeV1 (see [5] Prop .. 4o8).

Nm.; d(-p,q)

=

d(p,-g) and hence f(p)

=

f(-p) and thus each

e

_n

is symmetrico Since (aeV1)cr is Hausdorff each en is structur- ally closed, and Un

=

[p E o e V1 : f(p) <

~}

structurally openo Now Nn oeV1 ^r:: ~ un and hence Nn

c

8V1 is a G_{0 •}

3.

THE SPLITTING PROPERTY

Let X be compact Hausdorff, a

regular Borel measure on X with ~( {x_{0 })}

=

0.. Define V

S

C(X) by

V

=

{f E C(X): f(x_{0 )} =~(f)}

Then V is a Banach space, ~d it is possible by using [11] Theorem 6.17 and Lemma 8 here to prove the following Proposition:

PROPOSITION 13. Let V be as above and J c V a closed subspace ..

Then J is an M-ideal if and only if J = JF for some closed F c X where x₀ ¢. F , or x₀ E F and supp ~ '- F contains at most one pointa

COROLLARY 14. (deV1)cr is homeomorphic to the space where all the sets F

n y '

F closed in X ' and ^X

0

Y = X" {x }

0

~ F or ( {x } U supp ~) c F form the closed sets of the topology on Yo

0 -

PROOF. If supp ~ \ F contains just one point x , then F' ⁼ F U {x}

is closed and JF

=

JF, ..

COROLlARY 15., .Assume supp ~ contains more than one point.. Then (deV1)cr has the splitting property if and only if

and (d₈V1)cr is never Hausdorff.

x ~ supp ~ ,

0

PROOFo If x₀ ~ supp ~ then it is simple verification to show that Y (defined in Corollary 14) has the splitting property. If x₀ E supp ^1.1. then i t is impossible to split the closed set supp ~ ..

(We all the time assume ~-tC {x_{0 })}

=

0 ) o Y is never Hausdorff since i t is impossible to separate the points of supp t-t ..

From Corollary 15 and Corollary 7 we have that

V = [f E C(X) : f(x_{0 )} =~(f)} is isometric to a predual L1 ^(~)-space if x₀

I.

supp 1-l .. This is also true when x₀ E supp 1-l • A more general result was proved by Gleit [10] and Bednar and Lacey [L.J.] ..

They proved if for each i

=

1 ,2, .... ,n, measure on X , III-li

II _::

1 , xi E X and

1-1· is a regular Borel

~

11-li

I (

[x1 , x 2 , ... , ^{xn} )}

=

0 , then V = {fEC(X) :f(x.)=f.l.(f), i=1, ... ,n} is isometric to a

~ ~

predual L1 (f.l)-space and a simplex space if all 1-l· are positive ..

~

Lima has pointed out that the condition

I

~

· I (

{x1 , .... , x } ) = 0 ,

1 n

i = 1, • ., ^o ,n , is not necessary for the conclusion that V is iso- metric to a pr~dual L1 (f.l)-space.

Prop ..

13

many measures

can now in a natural way be extended to finitely 1-1 • ⁰

~

EXAMPLE

1. Let m be the Lebesgue measure on [0,1] and

V = {f EC([0,1]): f(t) =m(f)}, (oeV1)cr does not have the splitting property since t E [0,1] = suppm.. Perdrizet [14] used this space as an example of a simplex space with a family of M-ideals such

1 1

that the intersect~on is not an M-ideal. Let Fn = {~ + n} , n =

2,3, .... ' then by Prop. 13 is an M-ideal for each n, but

i

E F

n

JF

=

JF where is not an M-ideal since n

and [ 0 , 1 ] '\. F

=

supp m ".. F contains more than one point.

EXAMPLE 2 .. Let 1-l =

fe:

₀ +te: 1 , and V ⁼ [fEC([0,1]): f(t) =~(f)},

(a e V1) cr has the splitting property since t ~ {0, 1} = supp 1J. ^o Let Fn' n=3,4, ....

ideal for each as above.

be as above, then by Prop.

13

JF is an M- n

n , but

n

J F is not an M-ideal by the same reason n

EXAMPLE 3. Alfsen gives in [1] Prop ..

:n:

7.17 an example of a prime simplex .. If 1-l is positive, ll~-tll

=

^{1 and ~-t({x} ₀ ^{})= 0 then}

V = {f E C(X): f(x_{0 )} = IJ.(f)) is a simplex space with nnit, and hence K

=

{p ^E V* : p positive and

liPII =

¹

J

is a simplex. Assume supp 1J. =X.

From Corollary

15

and an earlier remark we have that K is prime.

Alfsen used in his example X

=

N U {co) (the one point compactifi- cation of the natural numbers), IJ.

=

2:: 2-n e: n and x o =co. . Now oeK with the facial topology is homeomorphic to (oeV~)cr and hence by Cor'Jllary

14

homeomorphic to

N

with closed sets

N, 0

and the finite ones (because a closed infinite subset of X must contain co) •

REFERENCES

1o E. Alfsen, Compact convex sets and bonndary integrals,

Ergebnisse math. Grenzgebiete, Bd.

57,

Springer-Verlag,

1971.

2. E. Alfsen and T.B. Andersen, Split faces of compact convex sets, Proc. London Math. Soc.

(3) 21 (1970), 415-442.

3.

E. Alfsen and E. Effros, Structure in real Banach spaces, part I ^& li, Ann. of Math.,

96 ( 1972), 98-173.

4. J.B. Bednar and H.E. Lacey, Concerning Banach spaces whose duals are abstract L-spaces, Pacific J. Math.,

41 (1972), 13-24.

5.

E. Effros, On a class of real Banach spaces, Israel J. Math.,

9 (1971), 430-458.

6.

E. Effros, Structure in simplexes, Acta Math.,

117 (1967), 103-121"

7.

H. Fakhoury, Preduaux de L-espace, Notion de Centre, J. Fnncto Anal.,

9 (1972), 189-207.

8. A. Gleit, Topologies on the extreme points of compact convex sets, part I, Math. Scan.,

31 ( 1972), 209-219 ..

9.

A. Gleit, A characterization of M-spaces in the class of separable simplex spaces, Trans. of AMS,

169 (1972), 25-33.

10.

A. Gleit, On the existence of simplex spaces, Israel J. Math.,

9 (1971), 199-209.

11. A.

Lima, Intersection properties of balls and subspaces in Banach spaces, Trans of AMS,

227 (1977), 1-62.

12. A.

Lima, On continuous convex functions and splitfaces, London Math. Soc., Proc~,

25 (1972), 27-40.

13.

J. Linn.enstrauss, Extensions of compact operators, Memoirs of AMS, 48

(1964).

14o

F. Perdrizet, Espaces de Banach ordonne et ideaux, J. de Mathe,

49 (1970), 61-98.

15.

N. Roy, A characterization of square Banach spaces, Israel. J. Math.,

17 (1974), 142-148.

16.

P. Taylor, A charac!;erization of G-spaces, Israel. J. Math.,

10 (1971), 131-134.