COMPLEX PREDUALS OF L1 AND SUBSPACES OF 1n∞ (C)

- 1 -

J.g!~:g.ct_ion

In the present- paper we investigate the structure of complex preduals of L1 and the problems concerning norm prese~Jing exten- sions of compact operatorso Most of the results are known in the real case, but the complex case does not follow from these in a straightforward manner; in fact, in many respects the complex case is much more complicated and requires often different proofso Many of the proofs here give other methods to show the corresponding real resulto

We now wish to indicate in greater detail the arrangement and the result of this papero

In section 1 we start by investigating the structure of those finite dimensional spaces, which embed isometrically into ~ for some no This leads then up to the main reslut of the section, which states that if X is a complex predual of L1 and E1 ^~X, E2 S X are finite dimensional spaces, so that E1 embeds isometri- cally into

F c X with

~ for some E1 c F, F

n, then for every e: > 0 there is an isometric to ~ for sui table m and d(x,F) < e: for x ^E E2 , 1\x\1 ^~ 1 • The corresponding real result was proved by Lazar and Lindenstrauss (16]o The proof given here proviedes an alternative way of proving the real resulto The main brick in the proof is a complex version of the Lazar selection theo- rem, recently proved in [26]. The result is then used to give a pew and very short proof of the Hirsberg-Lazar characterization of pre- duals of L1 , whose unit ball contains an extreme pointo

Section 2 is devoted to the study of norm preserving exten- sions of compact operatorso We first prove that if E is a finite dimensional space, whose unit ball is the absolutely convex hull of

finitely many points, then every point in BE can be represented as a combination of extreme points so that the coordinate functions are continuous. The real case is due to Kalman [12]o While his proof is geometric, the proof here uses an argument on extension of operators, based on the main theorem of section 1, but in the real case we do not need this theoremo

/ The previous results of the paper is then used to characterize those preduals X of L1 with the property that every compact operator with image in X can be extended preserving the norm.

The corresponding real result was proved by Lazar and our proof

follows his ideas. We end the section by proving that every predual of 1₁ is isomorphic to a ~

001

^-space, a result due to W.B. Johson and the first named aurthor.

'

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0.. Notations and preliminaries

In this paper all Banach spaces are assumed to be complex spaces unless otherwise stated, and throughout the paper we shall use the notation and terminology commonly used in Banach space theory as it appears in [22], let us just here recall that if X and Y are Banach spaces, then the Banach distance d(X,Y) is defined by

d(X~ Y)

=

inf {1\S\l \1S-1

l\ I

S isomorphism of X onto Y]

and if X and Y are not isomorphic we put d(X, Y)

=

oo.,

For every natural number n we let (e.

11

^{< j < n]} denote the

J - -

unit vector basis of ln and (e": \1 < j < n} its biorthogonal system,

1 J - -

i .. e.. the unit vactor basis of ~.. Further we let T be the unit circle in a; , and if x and y are vectors in a Banach space, then we say that x and y are T-eq_ui valent , if there is a t E T so that x

=

ty,

If X is a Banach space we let Bx denote the unit ball of X, and for a convex set K c X we let oeK denote the extreme points ..

A compact absolutely convex set K c X is called a complex polytope, if there exists a finite set A co K

e so that oeK

=

T • A.

Let E and F be locally convex spaces and denote by c"(F) the set of all non-empty convex subsets of F" If KcE is convex and cp: K

-

c(F) , then cp is called convex, provided:

for all x1 ,x2 E K and ^A. E [0,1] ..

The map cp is said to be lower semicontinuous if

{x E K

I

cp(x)

n

U

.P

¢} is open for every open subset U of F ..

Finally when K is absolutely convex, we say that cp is T-~­

metric, if cp( tx)

=

tcp(x) for all t E T and x E K ^o By a

selection for ~ we mean a map f: K- F such that f(x) E ~(x)

for all x E K.. Else our general reference in convexity is Alfsen' s book [1] ^o

1.. Structure theorems for preduals of _{L 1}

Before we prove the main theorems of this section mentioned in the introduction, we need the following easy proposition on complex polytopes:

1 .. 1. PROPOSITION. ^~ E be a finite dimensional Banach space, then ~ is a complex polytope, if and only if there is a n E 1N and an operator

the unit ball of E ..

taking the unit ball of ^{l n} ₁ .2!W2. ^t

Proof.. Assume first BE is a complex polytope, and let

x₁ ,~, o . . . ,xn be extreme points of BE, mutually non T-equivalent,

and so that oeBE

=

T" [x₁ ,x_{2 , ....} ,xn} .. If we define S: l~ ... E by:

S (

I:Z:

₁ t . e . ) =

I:Z:

₁ t . x . ,· t _{1 ,} t 2 , ... , t E C ,·

J= J J J= J J n

then it is obvious that S has the required properties ..

put If A

=

S: l~ - E satisfies the conditions in the proposition (ej

I

S(ej) E oeBE}. If ^X E oeBE' then

s-

^1(ej)

n

^B n is a compact face of B 11

ln and hence it contains an extreme

we

point, thus there is an 1 index j and t E T , so that S( te.)

=

x ,

J

but the.n e. E A,

J and x E T • S(A) • Hence o eBE ⁼ T " S(A) •

q .. e .. d.,

1 .. 2o COROLLARY.. Let E be a finite dimensional Banach space ..

Then E embeds isometrically into ~ for some n if and only if

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BE* is a complex polytope ..

We are now ready to state and prove the main theorem of this section.

1 ..

3.

THEOREM.. Let X be a predual of L1 and let F1 and

F2 be finite dimensional subspaces of X with _F1 isometric to a subspace of 1~ ^{for some k 0} Then for every e: > 0 there is a subspace E of X with _{F1 c} E, d(x,E) ~ e: for every

and so that E is isometric to ~ for sui table m •

Proof.. It is enough to prove the theorem in the case

dim _F2

=

1 , the general case will then follow by induction. Hence

let e: > 0, [y.

11

^{< i < n}} be a unit vector basis for F"' and z

J. - - I

a unit vector in _{F 2 •} We define R: BX* - ^(J) x (J)n by Rx* = (x*(z),x*(y₁),oeo,x*(yn)); x* E BX* ;

and put W

=

RBX* o Denote by D₈ the disc in <t with radius e:

and center 0 .. Since by our assumptions and corollary 1.2 the projection of W onto the last n coordinates is a complex poly- tope, and since oeW is totally bounded, we can find mutually non . T-equivalent extremepoints w₁,w₂,ooo'Wm of W so that if W'

=

conv (To {w_{1 , ....} ,wm}) , then

(1) (zE([)

I

(z,v)EW}

S

{zE<D

I

(z,v)EW'}+De Define S: 1 ~+^{1 .... <D} x a;n by

(2) S(ej)

=

wj; 1,::j~m S(em+~

=

(e:,O, .. ^eo ,0).

closed convex subsets of by:

(3) _W 1 (x*)

= s-

^{1 (Rx*)}

tkl

^{if Rx* =} twk, t E T w2(x*)

=

B m+1 else 11

w(x*)

=

^1j11(x*)

ⁿ

^1j12(x*)

It is easy to see that

w

is convex and T-symmetric. We wish to show that

w

is lower semicontinuous when BX* is equipped with the w*-topology. If U is an open subset of 1~+¹ , then the sets A.

=

{t E T

I

te. Ji! U} , 1 ~

j.:;:

m are compact and by the open mapping

J J

R-1

csu)

is w*-open subset of BX* ; therefore the set

is w*-open in BX* • This proves that $ is lower semicontinuous.

By the complex analogy of Lazar's selection theorem [26, theorem4.~

1jl admits an affine, T-symmetric and w*-continuous selection ~ o For k

=

1,2,.o.,m+1 we define xk EX by

Let a.1 , ^c.2 , .•• , a.m E ^{<V •} By the definition of \jr 2 we now get:

II

~=1 a.i xi

II =

^{sup {x*}

(t:l=1

^{a.i xi)}

I

x* E BX*)

= sup{q=1 "i er (~(x*))

I

x* E Bx*}

= sup{~ 1 ^Q.. e'!' (e)

I

^{e E B }}

~= ~ ~ lr~

=

sup

{I

a..

I I

1 < i < m}

]_

- ^-

This gives that the linear span of (xk}~=

1

is isometric to ~ • By (4) and the definition of \jl1 we get for all x* ^E BX* :

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If we put coordinatewise

(6) and

so the proof is complete.

As a corollary of the above theorem we can take out the next result, proved in the real case by Lazar and Lindenstrauss [16] ..

This is a slightly stronger version of the main result of Michael and Pelezynski in [23] ..

1 .. 4 .. THEOREM.. Let X be a separable predual of L1 and let F c X be a finite dimensional space which embeds isometrically into

I!

for some k .. Then there exists of finite dimensional subspaces of F c E1 , dim E 1 = 1 ⁺ dim E and

- n+ n -

an increasing sequence (En) co

X with X = U E and so that n=1 n dimE

En isometric to l co n ..

Proof. The result can be proved as in [16] by using our theorem 1 ^o 3 instead of their theorem 3 .. 1 ^o The fact that the E-d s can be chosen to satisfy dim E 1 ⁼ 1 + dim E

n+ n follows from [23] ..

We pass now to give an alternative proof of a functional repre- sentation theorem for complex preduals of L1 whose unit ball has an extreme point, due to Hirsberg and Lazar [10].. A simpler proof than the original one was given by Lima [17] ..

1 ..

5.

THEOREM. Let X be a predual of L_{1 ,} whose unit ball has an extreme point eo Let S

=

[x* EX*

l

x*(e)

=

1

=

1\x*l\

J

be

equipped with the w*-topology .. If ~:X- C(S) is the natural map defined by ~ (x) (x*)

=

x* (x) , x* E S , then ~ is an isometry, which maps X onto the space of w*-continuous complex affine func- tions on S ~ ~(e)

=

1.

Proof. Clearly ~(e)

=

1 and \l~(x)fl _::: llxll for all x E X ..

Let y EX with IIYil

=

^1. We wish to show that llw(y)\1 ~ 1 .. If

€ > 0 is arbitrary, then by theorem 1.3 we can find a finite di-

mensional space E ~X so that e E E and d(y,E) < ^€ and E isometric to

J!

for some n • Let (xj

)j

=1 be a basis for E isometrically equivalent to the unit vector basis of

J!

^{and let}

(x~)~ ^c x* be a sequence biorthogonal to J J=1 -

1 ~ j ~ n • By the above there is an x E E

(xj

)j ⁼

1 with llxj 11

=

1 , with llxll

=

1 and

IIY-xll _::: 2€.. Let j be chosen so that lxj(x)

I =

1 , since e is an extreme point of Bx

moreover:

\x":(e)l

=

¹

J and therefore x~(e)x": E S,

J J

lxj(e) xj(y)

l

~ \xj(x)

I -

llxjllllx-yl\ > 1-2e ,

hence 1\~(y)\1 _::: 1. An argument of [26] gives that $ is onto.

We want to thank A .. Lazar for suggesting this proof.

1.6 .. COROLLARY. ~ X be a predual of L₁ ~ e E X with lleiJ

=

^{1 o}

!J!i

^{S = {x* EX*}

l

^{x*(e) = 1}

=

^\lx*\1}. Then the following statements are equivalent:

(i) e is an extremepoint in Bx.

(ii)

s

is an maximal face

i£

Bx* ⁰

(iii) e considered as an element in BX** is an extreme point ..

~ 9 -

Proof. Assume (i) and that S is not a maximal face in Bx*o Then there exist y* E Bx* such that y* ¢ CoiiV [ tS

I

t E T} o By Hahn-Banach there exist a w* -continuous functional x , that is x E X , such that

x*(y) = 1 > sup{Re x(x*)

I

x* E

COiiV

[t S It E T}}

> sup{ lx*(x)

I I

x* E S}

which contradicts the fact that the map ~ in the preceeding theo- rem is an isometry.

(ii) => (iii)o Assume S is a maximal face in Bx* • We may identify. x* with L1 (Q,@,m) for some positive measure space

(Q,~ ,m) • First we observe that for any B E @ there is f E S with llf·xBII > 0.. If not the norm would be additive on the set

conv (S U {Ill(B)-1xB}) so by [2, lemma 2.1] we get a contradiction to the maximali ty of S .. Assume there is B E @ with m(B) > 0 and \e(q)l < 1 a. e. on B • By the above observation there is f E S with llf·xBII > 0. Since S is a face llt·xBII-1 f•XB

e.

S.

But \e(llf·xBII-1f·xB)

I

< 1 contradicting the definition of S.

(iii) => (i). Trivial.

Remarks. Functional representations of the tYI'e in theorem 1.5 where investigated and studied by Kadison (see [1, p.78]) who repre- sented the self adjoint part of a C*-algebra 0( isometrically as the real affine w*-continuous functions on the state space. In this situation one can no longer represent 0( isometrically as complex affine functions on the state space unless QC is commuta- tive. This is probably well known, but since we are unable to give a reference to this fact, we shall give a proof which was sho~~ to us by Christian Skau.

Let a E OL.. By assumption there is a pure state p such that \Ia\\

=

\p(a)

1 •

Let rrp be the corresponding representation with syclic vector s. Then we have 1\a\l = \p(a) \ = l<rrp(a)s,s) \

< l\rrp(a)s\\1\sll.:::, \\all. By Schwartz's equality rrp(a)s

=

p(a)s. If

b E ot, then p(ba) = (rrp(ba)s,s>

=

(rrp(b)rrp(a)s,s) ==

p(a)(rrp(b)s, s) = p(a)p(b). Similarly we get p(ab)

=

p(a)p(b), so p is multiplicativeo It follows that the spectral radius co- incides with the norm on 0( , so [4, theorem 4 .. 7] gives that 0( is commutative.. (The above result is incorrect for Banach algebras, consider the bounded operators on a predual of L₁ [4, p.87]) ..

On the other hand functional representations of Banach algebras will always be isomorphisms due to the Bohnenblust-Karlin theorem

[4], and for C*-algebras the onto argument is still valid, in fact this gives a characterization of the C*-algebras among the Banach algebras.. This is just a restatement of the Azimov-Ellis geometric interpretation of the Vidav-Palmer theorem [3]o

Let

ct

be a Banach algebra with unit and assume 0( is complex predual of L_{1 ..} Then the map ^1\f of theorem 1 .. 5 is onto, so ⁰⁽

is a C*-algebra. Since ~ is an isometry,

ot'

is commutative, so

~is isometric to C(S) for some compact Hausdorff space S ..

This result was proved by Hirsberg and Lazar for function algebras (10] and in general by Ellis (8] ..

2.. Norm preserving compact extensions

In the real theory of norm preserving extensions of compact operators the subspaces of the spaces ~ play a central role .. The --,....

/same is the case in the complex theory, as our results in section ¹

- 11 -

indicate; where there is one major differenceo In the real theory the subspaces of the ~'s are exactly the spaces, whose unit ball is a polytope (this follows for example from corollary 1 .. 2 and the fact that the unit ball of a real Banach space is a polytope if and only if the unit ball of the dual unit ball is a polytope [13]);

this is not so in the complex case as the example ~ shows.

We recall that a function f on a circled subset K of a Banach space is called T-homoheneous if f(tx)

=

tf(x) for all xEK, tET.

Before we can prove our main results we need the following:

2.1 .. PROPOSITION. Let K be a com12act Hausdorff s12ace,

x₀ ,x1 , .... ,xn E K •

'

A1 'A2' • • • 'An E ~ ; so that ^L:~ 1

I

A .

t

< 1 .. Then

J= J -

the subs12ace X of C(K) consisting of those f E C(K) for which f(x_{0 )} ^r:: l:~ _J= 1 A. f(x.) _{J J} is a 12redual of L1 ..

Proof .. We shall assume IAjl < 1 for all j , since else X is a G-space and hence is a predual of L1 [25].. It is immediate that oeBx* = {t l:lx l xEK, x;lx_{0 ,} t ET} .. Let cp: TxX _. oeBx* be the onto map defined by cp(t ,x)

=

t bx , t E T , x E K; and let IJ.

and v be two boundary measures on BX* with the same barycenter ..

According to a theorem of Effros (7], it is enough to show IJ.(f)

=

v(f) for all T-homogeneous f E C(Bx*) .. By the Hahn-Banach theo- rem there exist Radon probabilities ~-.~.1

cp(IJ.1) = IJ. , cp( v1 ) ⁼ v .. By maximali ty hence ~-.~.1 and v1 are concentrated on

and v1 on T x K so that IJ.({bx })

=

v(U>x }) = 0,

0 0

T ^X (K {x_{0 }) •} Let

f E C(Bx*) be T-homogeneous, and let e > 0. By regularity we can find a compact subset K1 E K ""- {x_{0 }}

that l~-.~.1 ^- v1¹ (T x K1 ) ^{~ l\1-.1.}1 - v1\\ - ^{e: •}

containing

By Tietze's extension theorem

we can find

f

^{E C(K)} so that f(x)

=

f( ox) , x _{E K1 ,}

Of'll

⁼ llfl\ ,

,..,

clearly f E X and hence

lllCf)- v(r)

l

~ I~Cr)- ^v(r)

I

+

1 J

^TxK (f( tox) - tf(x)

)d(~ 1 ^-v 1 ⁾

^{( t ,x)}

^l

~ J \reo ) - f(x)ldl~ 1 ^-v 1 ^{1Ct,x) ~}

21lf!l·e • Tx(K K1 ) x

Since e was arbitrary ~(f) = v(f).

We are now able to prove the following theorem on complex poly- topes.

2~2. THEOREM. ~ E be a finite dimensional Banach space whose unit ball is a complex polytope, and let x 1 ,x2 , ••• ,xn ^~ the extreme points modulo ^{T •} If x ^E BE with x = ~ ₁ A~ x. ,

- - 0 - 0 J= J J

~j

=

1 } Aj

I

~ 1 , then there exist functions A_{1 , A2} ^{,o • • ,} An E C(BE) ~ that ~r: ₁1 A.. (x) I < 1 , x

=

~ _{1 A.} (x)x. for all x E BE , ~

- J= J - J= J J

A.. (x )

=

A~ ^{, 1} < j

<:

n •

J 0 J - -

Proof. Let X be the subspace of C(~) consisting of those f for which f(x_{0 )}

= }:r: ₁

^A.~f(x.) and let S: 1n_{1 ...} E be the oper-

J= J J

ator from Proposition 1.1. Further let I: E* ... C(BE) be the cano- nical embedding; clearly I(E*) ~ X. Since E* is isometric to a subspace of ~ (viaS*) i t follows from theorem 1.3 and proposi- tion 2.1 that there is an m and a subspace F of X isometric to 1! with I(E*) ~ F.. Since F is a .. q;-space there is a norm one operator

I:

1~ ... F so that

If x E BE , then

,..,

I

=

IS* •

(1)

}:j= 1 \Aj(x)\ =

= 111*

oxfl n .:: 1 11

11 ~X: ₁ e ":

(I o )

e -II J= J x _{J 1}n

1

- 13 -

For all y* E E* we obtain:

y*(~ ^{/1 A.} .(x)x.)

=

(S*y*)(I*o )

J= 1 J J X

=

(IS* y*) (x) = y* (x) and hence

(2) x = ~ ₁ A.(x)x.

J= J J

2.3. COROLLARY. Under the same conditions as in 2.2 there exist T-homogeneous functions f₁,ooo'fn E C(BE) so that

X= L::r: _{J= 1} f. (x)x. and _{J J} ~ _{J= 1} lf . (x) _J

I

_-< 1 for all X E ~·

Proof. Let A. , 1..:::, j ~n be as in theorem 2.

J Define for each

= s

^{t -1} A . ( tx) d t , x E BE , T J

where dt is the normalized Haar measure on T • It is easily checked that the f.'s

J satisfy the requirements.

Remark. A slightly weaker form of corollary 2.3 was proved in the real case by Kalman (12] using geometric arguments. The real version of theorem 2.2 was proved by Lazar [15] by modification of Kalmans proof. Note that our proof of 2.2 and 2.3 with obvious

changes give an alternative proof in the real case, without using theorem 1.3, in fact it is easy to see that a C(K)-space has the finite binary intersection property, then argue as in [20, proof of theorem 5.5] ..

Recall that a linear subspace N of a Banach space W is called an L-ideal if there exist a subspace N' and W such that W == N$ N' and !lwll ""' llxll +

IIYII

for all w = (x,y) E NEB N' "

The reader is refered to [2], [9, theorem 1.2] and [17] for results on L-ideals.

The proof of the next proposition was suggested to us by

Ao

Lima.

2.4. PROPOSITION. Let (0.,(}3 ,ll) be a measure space and let F be a closed face in BL₁ (ll) and put E = span(F) • Then:

(i) E is an L-ideal (ii)

(iii)

E

n

_BL1 (ll)

=

conv (T·E)

1f

L1(1l) is a dual space and F is w*-closed, then E ~ w*-closed.

Proof. Since F is contained in a maximal proper face whose cone defines an order in L1 ^(ll) which makes it an abstract L-space, it is no loss of generality to assume that F is contained in the positive cone of L1(1l).

If we let C denote the cone generated by F , then it is readily verified that E ⁼ (C-C) ⁺ i(C-C) , and since C is heridi- tary [2, lemma 2.7] it follows that if f E E then f > 0 if and only if f E C • As a face cone in an L-space is a lattice cone we get by the above

( 1) E = {f E L1 ( 1-l)

1 If I

E c}

Since C is closed and the lattice operations are continuous, E is closed by (1). This relation also gives that E is an L1-space

- 15 -

under the induced ordero To prove (i) we first observe that by (1) E is a solid subspace of L

1

^(~) in the sense that f E E, lg} ~ If

I

implies g E E • Since E is an L1-space under the inducted order, monotone, norm bounded nets in E converges [24], [27], and hence E is a band in L1 ^(~) • It follows that E is an L-ideal.

(ii) will follow from

(2) BL1(~)

n

^{E = {f}

1

^{If\ E} conv(F,{O})} = conv(T .. F)

The first equality in (2) is obvious by (1). If f E L1^(~) with If

I

^E conv(F U (0})

g with

I

g I .::

t r

I

and _{e: > 0 '} then we can find a simple function

g

=

^{I: . -1} m ^{a., XA}

J- j

and llg-f\1 ^{< e: D} Hence g ^E E. Let

llxA

.11-¹

xA.

with ^{A •}

n

^{A .}

=

¢ ' i

F

j ' then J ^l.

J J

1 .:: j .:: n • Furthermore 1 ;:_

II

g\1

= Ej =

₁

l

a.j

Ill

^XA .11 and thus J

E F,

g E conv(T·F). The inclusion "CCriV(T.F) ~ BL

1

^(~)

n

E is trivial.

Finally assume that L1^(~) is a dual space and F is w*- closed. According to the Banach-Dieudonne theorem it is enough to prove that E

n

^BL₁^(~) is w*-closed. It follows immediately that C

n

_BL1 (~) is w*-compact. I f (ft) ~

En

_BL1 (~) is a w*-convergent net with limit f, then by a compactness argument we may assume

that the nets ((Reft)+), ((Reft)-), ((Imft)+) and ((Imft)-) all converges to elements in C

n

^BL₁^{(~) • Thus f E} E and tri vi- ally llfll .:: 1 ^D q.e.d.

The next lemma is actually one of the implications in our main theorem, but we have taken it out separately of technical reasons.

2.5. LEMMA. Let X be a predual of L1 •

!f.

^{BX* has an}

infinite dimensional w*-closed face, then X contains a subspace isometric to c •

Proof. Let F be an infinite dimensional w*-closed face of Bx* and put

L-ideal of X*

N

=

span(F) • By proposition 2 .. 4 N with F as a maximal proper face of

is a w*-closed

Z ⁼ X/N° then Z*

=

N, and since F is split in conv(FU -.iF) every f E A(F) (here A(F) denotes the complex, affine, w*-con- tinuous functions on F ) can be extended to an element in Z [25];

hence the map $: Z ~ A(F) defined by ($z)(x*)

=

x*(z) is an iso- metry onto. By Zippin's result [28] there is an isometric embedding

U: Re c ~ Re A(F) • I f W: c ~ A(F) . is defined by W(x+iy)

=

ux + i UY x,y E Re c then W is an isometric embedding.. In fact let s E F, with

II w

(X +iy) II

=

t ( ( u X) ( s) + i ( u y) ( s ) )

=

u ( u X) ( s ) - v ( u y) ( s)

= .u(Ret(x+iy))(s) ~II U(Ret(x+iy)!l ~ llx+iYII

In a similar manner we get 1\W(x+iy)!l ~ llx+iy!l .. (The last argument was shown to us by

A ..

Lima.) Hence we have shown that Z contains a subspace Y isometric to c •

I f (x ) c Y is a dense sequence, then we can define a metric

n -

on Y* with the aid of this sequence, so that it generates the w*- topology on By* .. Let us denote Y*'s completion in this metric

A

by Y*

'

clearly By* can be considered topologically as a subset of Y* .. ...

Let ^~: BN ^~ By* be defined by:

( ~ x* ) (y)

=

x* ( y) , y E Y , x* E N ^o

From [26, theorem 4 .. 2] there is an affine, T-symmetric w~continuous

map ~ : BX* ... By* , so that ~ IBN

=

^{cp ..} Define S: Y ... X by x*(Sy)

=

(~x*)(y), y E Y, x* E BX* ^o

Clearly S is an isometry and hence c embeds isometrically into X ..

q .. e.d.

- 17 -

We recall that a Banach space X is called an

;t

00,1 ^~space, if for every finite dimensional subspace E c X there is a finite dimensional subspace F c X with E c F and d(F,l~mF) ^< A • It is wellknown that a Banach space X is a predual of _L1 if and only if it is an _{: [00 1} +E: -space for all ^{E: >} 0 [22] .. The Banach space co is an

:£

00

'

1-space , whilw c is not, as it is seen from:

'

2.6. LEMMA.. There is a two dimensional subspace E of c , which does not embed isometrically into ,n _-'-00 f or any n.

P f L t ( k -1 )X> ( . -1 )'X)

roo .. e x₁

=

cos k=_{1 ,} x2

=

s~n k k=₁ and put

E

=

span x( ₁ ,x2 ) ^o F or he element xk t

=

cos k ^-1 x₁ + s~n ^. k ^-1 x2 ,

k E JN, we get xk(n) = cos(n-¹-k-^{1 ) ,} n,k E JN; and hence

~k) =

^{1 ,}

lxk(n)

I

^< 1 when n

I

k.. This shows that on E E* defined by

o

_n (x) = x(n) for all x E E , n E JN" is an extreme point ..

Corollary 1.2 now completes the proof ..

In [15] Lazar characterized those real Banach spaces X which have the property that every compact operator with image in X can be extended preserving the norm. A similar result is true in the complex case; the proof of it goes along the lines of [15, proof of theorem 3] ..

2 .. 7 .. THEOREM.. ~ X be a 12redual of L_{1 •} The following statements are equivalent:

(i) X

-

is a

:i

00

_'

1-space

(ii) No subspace of X is isometric to c

(iii) Bx* has no infinite dimensional •tJ"*-closed faces

(iv) For all Banach spaces Y ~ Z with Y ~ Z and every compact operator S: Y ... X , there is a compact extension

S: z ...

^X

!!i!h IISI\ =

^IIBII

(v) For all Banach spaces Y and Z with Y ~ Z and every operator S: Y ... X ~ dim S(Y) ~ 2 , there is a compact extension

S:

^{Z ....} X with liS!! =

I!Sll •

Proof. (i) ~ (ii): follows from lemma 2 .. 6 (ii) ~ (iii): is lemma 2o5

(iii) ~ (iv): Assume that (iii) holds and let S: Y ... X be compact with IISII

=

^{1 •} It follows that S* is continuous from BX*

to By* , when the first ball is equipped with the w* -topology and the latter with the norm topology. We wish to construct an affine, T-symmetric map ~= BX* ....

By• ,

continuous when the sets are equipped with the w*-topology, respectively the norm topology, so that

~x*

I

^Y

⁼

S*x* for all x* E BX*

Put K = S*Bx*. Arguing as Lazar [15, p.360] we get that there are finitely many non T-equivalent extreme points of K such that

We also get that there is a ~ O<S<1, so that if y* E oeK with IIY*II ^> ~

'

then IIY*

II

^{= 1 0 Put}

( 1) Kj3 ^{= {y*} E K

I

^I!Y* II~~}

The Krein-Milman theorem gives that (2) K = conv(K13 U To {u1, o o o , u~})

and

(3) K n {y* e Y*

I

^\ly*

II =

¹

1

^c conv(To {u1, ^{o • • ,}

u;;_}) =

K1

- 19 -

Let for z* E Z* ₁ be the Hahn-Banach extensions of u": .. We define a map w of K into the closed convex subsets of

J

Bz* by

(4) w(y*) = [z* EBZ*

I

z*lY=y*} for y* E K and 1\y*ll < 1 (5) w(y*) = {~ 1A .z":

I

y* = ~ 1A .u":' ~ 11A .{ = 1} for y* E K'

J= J J J= J J J= . J

IIY*II =

^{1 •}

Clearly w(y*)

I ¢

for all y* E K and it is readily verified that

*

is convex and T-symmetric. We shall prove w is lower semicontinuous, when K and Bz* are equipped with the norm topo- logies. Hence let U be an open subset of Z* and let

y; E (y* E K 1 w (y*)

n u 1 01 =

~

•

If

1\Y*ll

< 1 , then we argue like

0

Lazar [15] to get that y*

0 is an interior point of K_{2 •} suppose that

II

Yo*

II =

1 and let z ₀ E w _. (y*) ₀

n u

with zo = where y* = I:~ ¹ A~ u ": and I:~ ¹

I

A~

I =

^{1 ..} By theorem 2 a 2

0 J= J J J= J

Next

~1A~z":,

J= J J

there are functions A1 ,A2 , ••• ,>..n E C(K1 )

Ej=1 1>..j(y*)\ ~ 1 for all y* E K1

so that y* = Ej=₁ >..j(y* )uj , and ^{A •} (y* )

=

A~ ^{, 1} < J. < n •

J ⁰ J - -

Let e > 0 be given so that the ball with center in z*

0

is contained in U , and let

w

₁ be a neighborhood of that

(6) \1~ J= _{1 A} J .(y*)z":- z*ll J 0 -< 3-¹ € for y* E W₁

n

K1

It is easy to see that there is a neighborhood

w

of y*

0

if y*

e wn

K and y* = o.y1 +(¹-o.)y2' where Y1 ^{E KS ,}

and radius y* 0 so

so that Y2 E K1 ' and A E [0,1]

'

^{then a. < 3-1€ and Y2 E W1 • Let y* E W, y* =}

(7) v* ⁼

with Y1 E KS , Y2 E K₁ a. z* + ( 1-a.) E~ 1 A. _(y2*) z":

J= J J

and let z* E w (y~p •

By the convexity of $ is v* E ¢(y*) and furthermore

Put

(8) .llz;-v*ll _:: a.llz;-z*ll + (1-o.)I!Lj"=1 A.j(y2) zj-z0ll

~ 2/3€ + 1/3€

=

€

so v* E ¹¹¹ (y*)

n

^{U ,} which gives that y*

0 is an interior point of

K2 , and thus we have proved that $ is lower semicontinuous ..

The map w ^o T* is w*-lower semicontinuous and T-symmetric and therefore by [26] it has a w*-contunuous affine selection ~·

Define

S: Z

~ X by

(9) x*(Sz)

=

~(x* )(z) , z E Z, x* E BX* ..

By the properties of ~,

S

is compact and it is an extension of S with

11811 =

1 ..

(iv) ~ (i). The proof of this implication is essentially the same as the proof of theorem 7.9 in [20], but let us give it for the sake completeness. Put for every n EJN

s = n-1 o::n sk)

n k=o

By the ergodic theorem on compact operators [6,p .. 711] Sn converges to a finite dimensional projection

P

with

IIPII

= 1 and has image

(10) F = (x EX

l

Sx = x}

Since X** is a

~-space

is a ..'l\-space and hence

and

F is

P* *(X**)

=

F , it follows that F isometric to I!imF • Clearly E c F ..

-

(iv) ~ (v): is trivial~

(v) ~ (ii): Assuming (v) the proof of the implication

(iv) ~ (i) shows that every two dimensional subspace of X embeds isometrically into ~ for suitable n, and hence according to

X does not contain c isometrically. q.e .. d ..

- 21 -

Remark. We do not know whether the condition (v) implies that X is a predual of L1 • Lima [ 17, theorem 4.10] proved that the answer is positive for real spaces, and in [18, theorem 4.1] he proved that the answer is positive in the complex case if we in (v) require dim S(Y) ~ 3 instead of dim S(Y) ~ 2 •

As a corollary to theorem 2.7 we get as in the real case:

2.8. THEOREM.

1f.

X is an Zco 1-space, then X*

=

11^(r) for some

r. '

Proof. Assume first ahat X is separable. If oeBX* is uncountable modulo T , then o eBX* has an infinite compact subset E (one may even choose E to be the Cantor set) with E

n

^tE

⁼ ¢

for all t E T '\ [1} • By [25, lemma 22] F = COiiV(E) is a \•T*-

closed face of BX* which contradicts theorem 2.8. Hence oeBX*

is countable modulo T and thus X*

=

11 •

The general case follows from this together with [14, theorem6, p.227] (the implication we need is also proved for the complex case, although this is not stated explisitly; it is also likely, using the result of [26], that this theorem carries over to the complex case).

The final result of this section due to W.B. Johnson and the first named author shows that every predual of _{1 1} is isomorfic to

:i::o

1-space.

'

2.9. THEOREM. Let X be a real or complex Banach space with X*

=

11 • Then there exists a

;t::o

_1-space ^Y which is isomorphic to X.

'

Proof. Let (x*) c X* be a basis isometrically equivalent to n -

the unit vector basis of 1 1 ^o Put for every natural number n En= span{x1, ••• ,x~} and define a new norm on X* by:

( 1)

Ill

x*lll = llx*

II

+ I::1 2-n d(x"' ,En) , x* E X"' (This renorming technique was used in [5].) Since the finite dimensional the unit ball determined by

Ill

^{Ill is}

E 's are _n w*-closed and hence

Ill

Ill is the dual norm of a norm Ill Ill on X which is readily seen to be equivalent to 11 II • Put Y

=

(X,

Ill liD •

We

shall show that Y is a teo _1-space. Put for every natural number n

y~

=

!llx111-

¹

x~

and let k

'

be a natural number and t 1 , t2 ,...., tk scalars. Then

(2)

Ill ~=1

^tn

Y~lll

^{= Ill}

~=1

^tn

Ill x~IJI x~lll

⁼

I:k 2-1 1t

I

(1-2-n)-1 + ~-¹ 2-n(I:~ 2-1 1t .1 (1-2-j )-1 )

n=1 · n n=1 J=n+1 J

= 2-1 I:k. I:j-1 2-nlt

.I

(1-2-Y)-1 = ~k

It I

J=1 n=o J o(.,j=1 j

which shows Y* = 11 ^o

If we shows that every w*-limit point of the sequence (y*) n has norm strictly less than 1 , then it will follow from theorem 2.7 for the complex case and Lazar [15, theorem 3] for the real case that Y is an :feo _1-space. Hence let x* E X"' and let (y * )

' "' nk

be a sequence with y"'

!:!....>

x* • Since

Ill

y*

Ill -

2 we get that

nk nk

llx*ll ~ 2-1 and therefore for n sufficiently large d(x* ,En)< 2-1 ^o This gives

q.e.d.

- 23 -

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I

&

II, Ann.

of

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3.

L.A. Azimov and A.J. Ellis, On hermitian functionals on unital Banach algebras, Bull. London. Math. Soc. 4 (1973), 333-336.

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1-,

13a

V. Klee, Polyhedral sections of convex bodies, Acta Math.

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H.E. Lacey, The isometric theory of classical Banach spaces

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Springer Verlag, Berlin Heidelberg New York,

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16.

A. Lazar and J. Lindenstrauss, Banach spaces whose duals are L

1

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Math. Soc.

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1£

^~p-spaces and their applications, Studia Math.

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J. Lindenstrauss and L. Tzafriri, Classical Banach spaces

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E. Michael and A. Pelczynski, Separable Banach spaces which admit ~approximations, Israel Jo Math.

4 (1966), 189-198.

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- 25 -

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UNIVERSITY OF ODENSE, DENMARK, AND

UNIVERSITY OF OSLO, NORWAY.