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Duality and Geometry of Spaces of Compact Operators
59
DUALITY AND GEOMETRY OF SPACES OF COMPACT OPERATORS Wolfgang Ruess * ) Fachbereich Mathematik Universitat Essen Universitatsstr.3, D-43 Essen Federal Republic of Germany
INTRODUCTION
The object of this paper is to discuss the problem of how - for given Banach spaces X and Y - the space K(X,Y) of compact linear operators from X into Y and its dual space reflect the geometric and topological properties of X and Y and their respective duals. Since X* and Y are closed linear subspaces of K(X,Y), they inherit trivially properties such as non-containment of Z 1 , the dual having the Radon-Nikodym property, weak sequential completeness, or reflexivity, from K(X,Y). The natural question thus is to find out which additional conditions are needed for the reverse implications: how can geometric and topological properties of K(X,Y) and its dual be recovered from the corresponding properties of (the presumably well known spaces) X and Y and their duals? Besides the properties already mentioned, we shall consider the representation of the dual of K(X,Y), and discuss how the various classes of extreme points of the dual unit ball of K(X,Y) can be represented by the corresponding classes of points in the (bi-) duals of X and Y. The paper is essentially written in survey form, but, except for section 5, the main stream results are presented with proofs. The spectrum of results considered here will cover only a very small sector of the subject matter and is restricted to those parts that I have been involved with during the past few years. PRELIMINARIES 1.1 SPACES OF COMPACT OPERATORS
1.
Our discussion is placed in the general context of the operator space Kw*(X*,Y) of compact and weak*-weakly continuous linear operators from X* into Y, endowed with the usual operator norm. This space, originally introduced by L. Schwartz [ 6 0 ] in 1957 as the so-called &-product XEY of X and Y , apparently h a s gone out of sight over the years. However, it has the advantage over K(X,Y) that, as far as methods of proofs are concerned, it is conceptually easier to handle than K(X,Y) itself, and that it comprises not only spa_es of the type K(X,Y) but also completed injective tensor products X O E Y , and thus many more concrete spaces of analysis particularly spaces of vectorvalued continuous functions and of vector-valued measures. More specifically, we have the following (well known) fundamental isometrical isomorphisms and isometrical embeddings, respectively:
-
This paper is based on joint work with H . S . Collins, Baton Rouge (Louisiana) [4,51 and C.P. Stegall, Linz (Osterreich) [52,531.
*)
K(X,Y)
k
-
W. R i m s
= Kw,(X**,Y)
k**
X % , Y U K w * (X* , Y )
x8y-
{x*-(x*x)yl,
-
and
X2,Y
= K
W*
X o r Y has t h e approximation p r o p e r t y , c f .
(1.2.a)
C(K,X)
F
(1.2.b)
= K,,(x*,c(K))
[601.
= C ( K ) % ~ X
{ x * h x*F)
c c a ( C , X ) = Kw,(X*,ca(C)) = c a ( 1 ) S 0
(X*,Y) i f
( K compact Hausdorf f )
EX
X*@)
{X*
(Here, C i s a u - a l g e b r a (of s u b s e t s of a nonempty s e t Q ) , and c c a ( C , X ) d e n o t e s t h e s p a c e o f c o u n t a b l y a d d i t i v e m e a s u r e s from C i n t o X w i t h r e l a t i v e l y compact r a n g e , endowed w i t h t h e s e m i - v a r i a t i o n norm, c f .[18,51] .) D u a l i t y r e s u l t s f o r Kw,(X*,Y) t h L s o f t e n a l l o w immediate s p e c i a l i z a t i o n s t o b o t h K ( X , Y ) - s p a c e s and X@,Y-spaces, and t h i s i s t h e g e n e r a l approach t h a t w e u s e i n t h i s p a p e r . 1 . 2 NOTATION
B a s i c a l l y , w e f o l l o w t h e c l a s s i c a l n o t a t i o n of Dunford and Schwartz [ 1 2 ] . A s u b s e t A o f a Banach s p a c e X i s c a l l e d v e a k l y s e q u e n t i a l l y p r e c o m p a c t i f e v e r y s e q u e n c e i n A h a s a weak Cauchy s u b s e q u e n c e . The u n i t b a l l of a Banach s p a c e X i s d e n o t e d by BX, and i t s e x t r e m e p o i n t s by e x t B X . The s p a c e s of a l l bounded, weakly compact, c o m p a c t , and n u c l e a r o p e r a t o r s from X i n t o Y - X and Y Banach s p a c e s - a r e d e n o t e d by L(X,Y) , W ( X , Y ) , K ( X , Y ) , * a n d N ( X , Y ) , r e s p e c t i v e l y . I n a c c o r d a n c e w i t h o u r d e f i n i t i o n of Kw,(X , Y ) , w e d e n o t e by L w , ( X z , Y ) t h e s p a c e of weak*-weakly c o n t i n u o u s l i n e a r o p e r a t o r s from X i n t o Y . The t e r m Radon-Nikodym p r o p e r t y i s , a s u s u a l , a b b r e v i a t e d by R N P , and t h e t e r m [ m e t r i c ] a p p r o x i m a t i o n p r o p e r t y by [ m . ] a . p .
2 . EARLY RESULTS W e b r i e f l y r e c a l l s e v e r a l c l a s s i c a l r e s u l t s on t h e d u a l i t y and geometry of K ( X , Y ) i n o r d e r t o g i v e a f l a v o u r of t h e r e s u l t s t o b e
expected 2.1
-
and t h o s e n o t t o be e x p e c t e d .
Dunford
/ S c h a t t e n [ I l l , 1 9 4 6 , and S c h a t t e n [ 5 8 1 , 1950:
I c o n t a i n s a c o p y of co i s o r n e t r i c a Z 2 y . P 2 . 2 S c h a t t e n [ 5 9 ] , 1957: The u n i t b a l l of KI121 has no e x t r e m e
For 1 < p < m
, t h e space
X ( l
points. 2 . 3 C o r o l l a r y ( Dunford / S c h a t t e n ) : L e t H be an i n f i n i t e - d i m e n s i o n a l H i Z b e r t s p a c e . T h e n K ( H 1 i s n o t r e f l e x i v e , and n o t e v e n w e a k l y s e q u e n t i a l l y c o m p l e t e , n o r i s o m e t r i c t o a dual. s p a c e .
These a r e , i n a s e n s e , t h e n e g a t i v e r e s u l t s . The message i s t h a t o n e c a n n o t e x p e c t K ( X , Y ) t o b e " n i c e " p r o v i d e d t h a t o n l y X and Y are " n i c e " enough. On t h e c o n t r a r y , K ( X , Y ) may j u s t become t o o l a r g e .
61
Geometry o f operator spaces
In the other direction, there are the following classical positive results. 2.4 Pitt t451, 1936:
For 1 I p < q < m , a l L b o u n d e d l i n e a r o p e r a t o r s
L (lq, Zpi = Kilq, Lpl . Combined with Grothendieck’s result 2.6(b)(i) quoted below, this implies that f o r 1 $ p < g < - , t h e s p a c e K ( Z l i i s r e f l e z i v e . q’ P
from L
9
into l
are compact:
2.5 Grothendieck [19], 1955: j
:
x*%,Y* x*-*
-
C o n s i d e r t h e c a n o n i c a l ( L i n e a r ) map
(K,,(x*,Y))*
(h+(hx*,y*)}.
T h e n we h a v e :
is s u r j e c t i v e i f X* o r Y* has R N P . ( b ) j i s i n j e c t i v e if X* o r Y* h a s a . p . More s p e c i f i c a L l y : I f e i t h e r o f X* and Y * h a s a . p . , (a)
j
p : X* g,Y* x*w*
-
i s i n j e c t i v e , and if p i s i n j e c t i v e ,
B(X,Y)
z(x,Y)-
then the map
(x*x)(Y*Y)I
t h e n s o is j .
2.6 Corollary (Grothendieck [ 1 9 ] , 1955): ( a ) I f X** o r Y * h a s R N P , t h e n K(X,Y)* is a q u o t i e n t of X * * & Y * , and if, i n a d d i t i o n , e i t h e r of X * * and Y * h a s a . p . , t h e n we h a v e : K I X , Y I * = X**-@,Y* = N ( X * , Y * ) . ( b l Assume t h a t X and Y a r e r e f l e x i v e . T h e n we h a v e :
( i ) If L(X,Y) = K(X,Y/, t h e n K(X,YI
i s reflexive.
( i i l C o n v e r s e l y , if K(X,Y/ is r e f l e x i v e and p : X**ZnY*-
is i n j e c t i v e ,
then LIX,Yi
B(X*,Y)
= K(X,Y/.
We pause for a proof of Corollary 2.6, for these consequences of Grothendieck’s results 2.5 later on have often been reconsidered (see Remarks 2.7 below)
.
P r o o f of Corollary 2.6: Proposition (a) is - via the isometry ( 1 . 1 ) just a special case of 2.5. For a proof of proposition (b), assume now that X and Y are reflexive. Then, according to (a), we have A quick inspection K(X,Y)* = XSTY*/Q, so that K(X,Y)** = Q’cL(X,Y). of the corresponding isometrical embeddings reveals that K(X,Y) is reflexive in case L(X,Y) = K(X,Y). This proves proposition (b)(i). As for (b)(ii), we need only use 2.5 to conclude from K(X,Y)*=XG, Y* (and the assumption) that K(X,Y)=K(X,Y)**=(XG, Y*)*=L(X,Y).
2.7 Remarks: 1. Theorem 2.5 is possibly better known as the result on the coincidence of integral and nuclear operators in the
62
W. R u e s
presence of RNP. Grothendieck [_?9,1.54.2,Thm.8]did not explicitly state 2.5, but proved it for XQ9,Y under the assumpion that X is reB(X,Y) is flexive or has a separable dual, and that p : X* any*injective. In his proof, however, he only uses that, under the assumptions on X, the dual unit ball BX* has what he called the "Phillips property" ("proprigtg @ " ) [ 19,1.54.1 ,Def.6,p.104] which, in present day language, exactly is the RNP for X*. For an explicit proof of 2.5, see Diestel [7], Gil de Lamadrid [161, Schachermayer [571, and C. Stegall [66,Cor.71. 2 . The results of Corollary 2.6, only implicit in Grothendieck’s thesis [I91 - as indicated above - have explicitly been proved by various authors. Feder and Saphar [14,Thm.l] proved proposition (a), also relying heavily on Grothendieck’s work [191. Proposition (b) appears in various (partly weaker) versions in Ruckle [501, Holub [28,291, Kunas K. Jun [341, Kalton r35,section 2,Cor.21, and, in the form stated above, in Heinrich 1221.
3. Note that proposition 2.6(b)(ii) particularly implies Dunford / Schatten’s result that, for an infinite-dimensional Hilbert space HI K(H) is not reflexive. Historically, looking back, it is not too surprising that after the early results on spaces of operators on Banach spaces mentioned so far, not much happened in this direction for about a decade, and that it took the vivid revival of Banach space theory in the sixties to renew the interest in the geometric and topological structure also of operator spaces. This process did not start before the early seventies. A survey on a small part of it is the subject of the subsequent sections.
Note: It seems worth specifying the assertion of Theorem 2.5 in the sense of our general program - to determine K(X,Y) and its dual from the spaces X and Y and their duals: (a) If either of p*and Y* has RNP, then every T E K(X,Y)* has a ZI , and representation of the form T = XX.x**@y* , with 1 1 1 XI*)^ and ( ~ f bounded ) ~ sequences in X** and Y*, respectively, in the sense that Tk = thi(k**xr*,yf) for all k E K(X,Y). (b) If X* has RNP, then every Q E C(K,X)* (K compact Hausdorff) has and a representation of the form Q = .EXixfOui , with (Ai)iE Z,, bounded sequences in X* and M (K), respectively, in (xf) and (pi) the sense that
Q F = tAi(xfFdpi
for all FEC(K,X).
This is to be noted as the advantage over the case for general XI where the dual of C(K,X) can only be represented as a space of more general X*-valued measures, cf. Singer [631, Prolla [46,Ch.V.51 and Schmets [61,p.368-3771.
63
Geometry of operator spaces
3. REFLEXIVITY AND WEAKER NOTIONS We first recall the implications among the various weakenings of reflexivity.
Z*
RNP
/
Z
reflexive
Z $
-
Z
weakly sequentially complete
Bounded subsets of Z are weakly sequentially precompact.
l1
Any of these properties is being inherited by X and Y from the operator space Kw,(X*,Y). According to our general theme, we now turn to the discussion of which additional assumptions are required for the reverse implications. 3.1 REFLEXIVITY AND WEAK SEQUENTIAL COMPLETENESS Generally speaking (and modulo the a.p.) , for K(X,Y) to be reflexive or just weakly sequentially complete, all bounded operators from X into Y must necessarily be compact. More specifically, the following results hold.
3.1.1
Grothendieck [I91 (see 2.5 and 2.6 above):
Assume t h a t X and Y a r e r e f l e x i v e and t h a t e i t h e r o f X and Y has a . p . T h e n K(X,Y) is r e f l e x i v e if and o n l y if L(X,YI = K I X , Y / .
3.1.2 Theorem ( D.R. Lewis [37,Thm.2.1]):
( a ) Assume t h a t X and Y a r e w e a k l y s e q u e n t i a l l y c o m p l e t e , and t h a t Lw,(X*,YI
= Kw,(X*,Y).
Then Kw,IX*,Y/
is w e a k l y s e q u e n t i a L l y c o m p l e t e .
( b ) C o n v e r s e l y , a s s u m e t h a t Kw* IX*, Y ) is w e a k l y s e q u e n t i a L l y c o m p l e t e ,
and t h a t e i t h e r of X and Y has t h e m e t r i c a p p r o x i m a t i o n p r o p e r t y . T h e n X and Y a r e w e a k l y s e q u e n t i a l l y c o m p l e t e , and L w * I X * , Y )
= Ku,(X*,YI.
N o t e s : Proposition (a) as stated here is to be-found in [4,Thm.3.41. D.R. Lewis [37,Thm.2.1] proved it for the space X&,Y and thus needed the a.p. in one of the factors also for this part of the assertion. A special case of proposition (a) is the result of F. Lust [401.
What is behind Theorem 3.1.2 is the following general result.
3.1.2’ [4,Prop.3.11: Kw,(X*,YI i s w e a k l y s e q u e n t i a l l y c o m p l e t e if and o n Z y if (i) x and Y a r e w e a k l y s e q u e n t i a l l y c o m p l e t e , and (ii) Kw* (X*, Y ) is w e a k - o p e r a t o r - t o p o l o g y s e q u e n t i a l l y c l o s e d in Lw* tX*, Y). Thus, D.R. Lewis’ particular achievement - with his beautiful proof of proposition 3.1.2(b) in [371 - was to show that condition (ii) in 3.1.2’ together with the m.a.p. in one of the factors implies equality of Lw*(X*,Y) and Kw* ( X * , Y ) .
W. Rues
64
3.1.3 Corollary (D.R. Lewis [37,Cor.2.41, and [4,Thm.3.51):
( a ) A s s u m e t h a t X* and Y a r e w e a k l y s e q u e n t i a l l y c o m p l e t e and t h a t W(X,Yl
= K(X,Y/.
Then K ( X , Y I
i s weakly sequentiaZZy complete.
( b ) ConverseZy, assume t h a t K f X , Y ) i s w e a k l y s e q u e n t i a l l y c o m p l e t e and t h a t e i t h e r o f X* and Y has m . a . p . q u e n t i a l l y c o m p l e t e and W ( X , Y I
T h e n X* and Y a r e w e a k l y s e -
= K(X,Y).
This is just a special case of 3.1.2 by using the isometries K(X,Y) = Kw,(X**,Y) and W(X,Y) = Lw,(X**,Y). For further special cases (and extensions to particular locally convex spaces), the reader is referred to section 3 of [41, and to Tsitsas [70,711. Note, finally, that 3.1.3 implies Dunford / Schatten’s result that K(H) is not weakly sequentially complete if H is an infinite-dimensional Hilbert space. 3.2 THE DUAL HAVING THE RADON-NIKODYM PROPERTY In contrast with the situation for reflexivity, weak sequential completeness and non-containment of Z1 (3.3 below) - where additional assumptions are needed - the RNP for both X* and Y* completely determines the RNP for the dual of Kw* (X*,Y)
.
3.2.1 Theorem [52,Thm.1.9]:
T h e d u a l o f Kw*
X*,YI
h a s R N P i f and
onZy i f X* a n d Y* h a v e R N P .
3.2.2 Corollary [52,Cor.1.101: The d u a l o f K X,Yl o n l y i f X**
has R N P i f and
and Y * h a v e R N P .
A particular consequence of Theorem 3.2.1 is the result that - in
case either of X* and Y* has a.p. - the space N(X,Y*) = X*%?Y* of nuclear operators from X into Y * has RNP, provided that X* and Y * have RNP; for, under these assumptions, we have Kw,(X*,Y) = X5,Y
and
Kw,(X*,Y)* = X*Gi,Y* = N(X,Y*) (see 2.5 above). This special case of Theorem 3.2.1 is proved in Diestel / Uhl [IO,VIII.4,Thm.7,p.249], and it is their method of proof that we generally follow in our proof of Theorem 3.2.1. P r o o f of Theorem 3.2.1 :
We use the fact that
Z*
has RNP if and
only if every separable subspace of Z has a separable dual: Uhl, cf. [10,111.3,Cor.6,p.82] and Steqall 1 6 6 1 . From this equivalence, the necessity part of Theorem 3.2.1 follows trivially. NOW, assume that X* and Y* have RNP, and let A be a closed separable subspace of Kw* (X*,Y) Choose a dense sequence (hn)n in A, and a closed separable subspace Yo of Y such that imhncYo for all nElN. Accordingly, for
.
(h:)n c Kw* ( (Yo)* ,X), choose a closed separable subspace Xo of X such Thus, we can view the sequence (hn)n as a that imh,*CXo for all nElN
.
65
Geometry o f operator spaces
*,Yo).N o w , it i s g e n e r a l l y t r u e ( c f .
s u b s e t o f Kw* ( (X,)
M and N c l o s e d l i n e a r s u b s p a c e s o f X a n d Y ,
[ 601) t h a t f o r
respectively,
t h e space
Kw* (M*,N) i s ( i s o m e t r i c a l l y ) a c l o s e d l i n e a r s u b s p a c e o f Kw* (X*,Y) T h i s r e v e a l s t h a t A i s a c l o s e d l i n e a r s u b s p a c e of Kw* ((Xo)* , Y o ) . (Xo)* and (Yo)* a r e s e p a r a b l e ,
a s s u m p t i o n on X and Y , b o t h
.
By
so t h a t
(Xo)* gT (Yo)* i s s e p a r a b l e as w e l l . A c c o r d i n g t o Theorem 2.5 i n s e c t i o n 2 , t h e dual of
%* ( ( X o ) * ,Yo)
i s s e p a r a b l e . Consequently,
i s a q u o t i e n t of
l1
A s s u m e t h a t X and Y do n o t c o n t a i n 2 1
and t h a t e i t h e r o f X* and Y * has R N P . T h e n K m , l X * , Y I t h e isometry K ( X , Y )
= Kw+(X**,Y) o f
t h e f o l l o w i n g s p e c i a l i z a t i o n t o K(X,Y) 3.3.2
Corollary [4,Cor.1.12]:
11, and t h a t X**
(Yo)* , a n d t h u s
This completes t h e proof.
O F ( AN ISOMORPH O F )
3 . 3 . 1 Theorem [ 4 , T h m . 1 . 1 4 ] :
A s usual,
ST
t h e d u a l of t h e c l o s e d l i n e a r s u b s p a c e A
o f Kw,((Xo)* ,Yo) i s s e p a r a b l e a s w e l l . 3 . 3 NON-CONTAINMENT
(Xo)*
o r Y* h a s R N P .
.
d o e s n o t c o n t a i n 21.
( 1 . 1 ) above a l l o w s
Assume t h a t X* and Y do n o t c o n t a i n Then K l X , Y )
does n o t c o n t a i n l l .
N o t e s : 1 . The r e s u l t o f Theorem 3.3.1 h a s b e e n p r e s e n t e d by Heron C o l l i n s a t t h e 1980 C o n f e r e n c e on "Measure T h e o r y a n d i t s A p p l i c a t i o n s " a t N o r t h e r n I l l i n o i s U n i v e r s i t y , DeKalb, I l l i n o i s , a n d f i r s t a p p e a r e d i n o u r j o i n t announcement o f r e s u l t s of o u r p a p e r [ 4 1 i n t h e P r o c e e d i n g s o f t h a t c o n f e r e n c e i n 1981, e d i t e d by G . A . G o l d i n a n d R . W h e e l e r , p.187-192. I t i s i n t e r e s t i n g t o n o t e t h a t B.M. Makarov and V . G . Samarsk i i announced t h e a s s e r t i o n of C o r o l l a r y 3 . 3 . 2 i n [ 4 1 , T h m . 2 . 3 ] , and t h a t , i n t h e p e r i o d when o u r p a p e r [ 4 1 w a s i n p r i n t , C . Sam-uel's paper 1541 a p p e a r e d where h e p r o v e s 3.3.1 f o r t h e s u b s p a c e X a C Y. T h i s d e v e l o p m e n t may h e l p t o s u p p o r t o u r p o i n t - s p e c i f i e d i n s e c t i o n 1 . 1 t h a t t h e " r i g h t " s p a c e t o b e l o o k e d a t i s t h e s p a c e Kw* (X*,Y)
.
2 . F a k h o u r i [13,Thm.3] is separable.
proved 3.3.2
P r o o f o f Theorem 3 . 3 . 1 :
u n d e r t h e a s s u m p t i o n t h a t X**
We u s e R o s e n t h a l ' s f u n d a m e n t a l r e s u l t
(see R o s e n t h a l [491 and Ode11 / R o s e n t h a l [ 4 2 ] ) t h a t a Banach s p a c e Z does n o t c o n t a i n
( a n i s o m o r p h ) o f Z,
i f and o n l y i f t h e bounded s u b -
s e t s of 2 are w e a k l y s e q u e n t i a l l y p r e c o m p a c t . Assume t h a t X + Z 1 Kw,(X*,Y) = K,,(Y*,X).)
a n d t h a t Y* h a s RNP. Let
( N o t i c e t h e symmetry ( h n ) n b e a bounded s e q u e n c e i n Kw,(X*,Y).
Then t h e r e e x i s t s a s e p a r a b l e
( c l o s e d l i n e a r ) s u b s p a c e Yo of Y s u c h
t h a t imhncYo f o r a l l n€lN. According t o o u r assumption, parable,
see S t e g a l l [ 6 6 1 .
Let
(YE),
(Yo)* i s se-
b e a dense sequence i n (Yo)*. Then (hGy;)n i s a bounded s e q u e n c e i n X a n d t h u s , s i n c e X 4 Z l , h a s
a weak Cauchy s u b s e q u e n c e
(h;tkyl;)k. C o n s i d e r i n g now t h e s e q u e n c e
66
W. Rues
(h$ky;)k
in X, continuing inductively, and using a diagonal process,
we arrive at a subsequence (hnl)l of (hn)n such that (hzlyz)l is weakly Cauchy in X for all kE7N. Taking adjoints, we conclude that (hnlx*) 1 * ) )-Cauchy in Yo for all x*EX*. But, since (yi), is normis a in (yk k* , on bounded subsets of Y dense (Yo) the weak topology o(Yo, (Yo)*) 0'
of Yo and the topology u (Yo, (yZlk) coincide. Hence, (hnlx*)l is weakly Cauchy in Yo, and thus in Y, for all x* € X*. At this point, we use the isometrical embedding
-
Kw* (X*,Y)
C (BX*XBy*)
h {(x*,y*)-(hx*,y*)I , to conclude from the classical weak convergence result for C(K)-spaces that (hnl)
.
is weakly Cauchy in Kw* (X*,Y) This completes the proof.
Going back to Theorem 3 . 3 . 1 ,
made heavy use of Y* having RNP
and inspecting its proof
-
-
where we
the question arises naturally whether
the assumption of the RNP for either of the duals is only a technical matter that could possibly be dispensed with by just requiring X and Y not to contain l I . Too much hope in this direction is put to rest by the following example. 3.3.3
Example [52]: L e t J T d e n o t e t h e J a m e s T r e e s p a c e , [ 3 9 ] , but J T Z ; , J T d o e s c o n t a i n Z 1 .
cf.
[39].
Then J T $ I I
Proof:
JT is the dual of a (separable) Banach space B such that
B**/B = 1; = Hilbert space of the dimension of the continuum [ 3 9 1 .
Using the fact that projective tensor products "respect" quotients,
we conclude that JT*Zn JT* has l ; ~ T e as l ~ a quotient. NOW, the "diagonal" in Z;z, l ; is isometric to l 1 , which thus can be "lifted" isomorphically to JT* 3, JT* , cf. Pelczynski [ 43 ,Lemma 3.1 1. But JT* JT* is a closed linear subspace of (JTGEJT)* (see 2.5 in sec-
sTI
tion 2 ) , so that the dual of the separable space JT& JT contains I f . Using Pelczynski’s and Hagler’s results [43,201, we conclude that JT~, JT= z l . A combination of Corollary 3.3.2 with Theorem 2 . 5 yields the following connection between reflexivity and weak sequential completeness of K(X,Y).
67
Geometry o f operator spaces
L e t X and Y be r e -
3.3.4 Theorem ([4,Thm.3.91 and [5,Cor.3.31):
fZexive,
and c o n s i d e r t h e f o l l o w i n g s t a t e m e n t s :
(a) LJX,Y) (el K(X,Y/
= K(X,Yl; f b ) L(X,Y) i s reflexive; i s r e f l e x i v e ; ( d ) KlX,Y) i s w e a k l y s e q u e n t i a l l y c o m p l e t e ;
T h e n ( a ) i m p l i e s ( b ) , ( b ) i m p l i e s ( e l , and (c) i s e q u i v a l e n t t o i d ) . Pinally,
i f t h e n a t u r a l map
j : x~~~Y*-(K(x,YJ)* i s i n j e c t i v e ,
t h e n ( d ) i m p l i e s (a). j is i n j e c t i v e , w h e n e v e r X o r Y h a s a . p .
For a description of general weakly sequentially precompact subsets of Kw* (X*,Y) via the corresponding concepts in the factor spaces X and Y, the reader is referred to Lewis [37,Thm.3.11, and to [4,section 1 1 .
4. WEAK COMPACTNESS IN K(X,Y), AND EXTREME POINTS IN THE DUAL OF K(XtY)
4.1 WEAK COMPACTNESS W e start with the well-known fact that weak sequential convergence and weak compactness are determined by convergence not necessarily on all elements of the dual but just on the extreme points of the dual unit ball.
4.1.1 Rainwater [47]:
A bounded sequence
Z c o n v e r g e s w e a k l y t o z E 2 i f and o n l y i f
( z n l n i n a Banach s p a c e
(z*z
f o r a l l z*EextBZ*.
)
n n
converges t o z*z
This result was the starting point for an intense study of the reduction of weak sequential convergence and weak compactness in Banach and more general locally convex - spaces Z to the corresponding notions with respect to the weak topology o(Z,extBZ*) on Z generated by just the extreme points of B *. W e refer the reader to K. Floret’s detailed exposition of this devefopment in [ 151. For our purpose here, it suffices to recall one of the latest results in this direction. 4.1.2 Bourgain / Talagrand [2]: A b o u n d e d s u b s e t H o f a Banach s p a c e Z is w e a k l y r e l a t i v e l y c o m p a c t i f and o n l y i f i t i s r e l a t i v e l y c o u n t a b l y compact w i t h r e s p e c t t o o ( Z , e x t B Z * l .
Note that, in result 4.1.2, the improvement over RainR e m a r k s : 1. water’s result - for describing weak relative compactness - lies in the reduction to o(Z,extBZ+)-relative countable compactness as opposed to o(Z,extBZ*)-relative sequential compactness. 2. There exist by now further proofs and extensions of Theorem 4.1.2 by Khurana [36] and Tweddle [731, a further simplification of Khurana’s proof by I. Namioka (personal communication), and a deduction of result 4.1.2 from Ramsey-type theorems by C. Stegall (oral communication).
68
W. Ruess
A l s o , it o u g h t t o be n o t e d t h a t f o r convex s u b s e t s H , b e e n known, see D e Wilde [ 6 1 .
4.1.2
has
According t o t h e d i s c u s s i o n s o f a r , one way t o c h a r a c t e r i z e weak c o m p a c t n e s s i n K *(X*,Y) i s t o d e t e r m i n e t h e form of t h e e x t r e m e p o i n t s i n t h e d u a l u n i t W b a l l of %* (X* , Y ) and t h e n t o a p p l y 4 . 1 . 1 and 4 . 1 . 2 . T h i s a p p r o a c h and t h e s u b s e q u e n t r e s u l t s of s e c t i o n 4 . 1 a r e t o b e f o u n d i n F l o r e t [ 1 5 , s e c t i o n s 8.10-8.131, and i n [ 4 , s e c t i o n 21. 4.1.3 Observation:
,*my*
extB (Kw*(X*, y I I * c fextBX*l@ (extBy*)
,
where
a c t s on hEKw*(X*,Y) i n t h e c a n o n i c a l way: x * @ y * ( h i = f h x * , y * ) . Kw* (X*,Y) i s a c l o s e d l i n e a r s u b s p a c e of C(BX*XBy*), BX*
Proof:
-
and BY* b e i n g endowed w i t h t h e i r r e s p e c t i v e w e a k * - t o p o l o g i e s :
-
Kw,(X*,Y)
'
C(BX*XBy*) I(x*,y*)t-t(hx*,y*)).
h
* i s o f t h e form (Kw* (X*,Y)) w i t h (x*,y*)EBX*xBy*. However, i f w e as-
Thus, e v e r y e x t r e m e p o i n t of B 6 ( x * , y * ) IKw,(X*,Y)
sume x***
=
x*@*
t o be an e x t r e m e p o i n t of B
t h e n , by t h e b i (Kw*(X* , Y ) )* l i n e a r i t y of t h e map (x*,y*)x*@y*, x* and y* n e c e s s a r i l y must b e e x t r e m e p o i n t s of B * a n d By*, r e s p e c t i v e y . T h i s p r o v e s t h e asX sertion.
I t i s now e a s y t o a r r i v e a t t h e f o l l o w i n g weak c o m p a c t n e s s r e s u l t s , c f . F l o r e t [ 1 5 , 8 . 1 0 - 8 . 1 3 1 , and [ 4 , s e c t i o n 2
4.1.4
(a)
Theorem:
A bounded sequence
hEKw,(X*,YI
i n Kw,(X*,Y)
i f and onZy i f ( h n x * , y * ) ,
(x*, y * ) E e x t B X , x e x t B Y *.
( b ) A bounded s u b s e t H o f K w * ( X * , Y )
eonverges weakly t o
c o n v e r g e s t o ( h x * , y * ) for a l l
i s weakly r e l a t i v e l y compact i f
and o n l y i f i t i s e x t B X * x e x t B * - w e a k - o p e r a t o r - t o p o l o g y
Y
countably compact.
relatively
W e n o t e t w o p a r t i c u l a r cases.
4.1.5
Corollary:
( a ) A bounded s e q u e n c e
( k n j n i n K(X,Y)
converges weakly t o k€K(X,Y)
if and o n l y i f ( k ; * x * * , y * ) n c o n v e r g e s t o ( k * * x * * , y * ) (x**, y*)EextB **xextBy*. X ( b ) A bounded s e q u e n c e weakZy t o F E C ( K , X ) F ( t ) for a l l t E K .
(Fnjn
i n CIK,XI,
for all
K compact H a u s d o r f f ,
i f and o n l y i f ( F n ( t ) ) n
converges weakly
converges ( i n Xl t o
Geometry o f operator spaces
69
For related weak compactness results, compare Kalton 135,section 21, and Tsitsas [711. After all, we can summarize that the reduction to the o(Z,extB * ) topology actually led to the desired description of weak compactngss properties by means of the corresponding pointwise-weak compactness properties. Finally, we derive a further result on the dual of Kw,(X*,Y) by combining observation 4.1.3 with a result of Haydon [21,Prop.3.1]. 4.1.6 Theorem ([52,Thm.1.7]): L e t H be a c l o s e d l i n e a r s u b s p a c e of Kw,(X*,Y), c o n t a i n i n g X s , Y . A s s u m e t h a t H d o e s n o t c o n t a i n Z 1 . T h e n H* i s a q u o t i e n t o f X * S T Y * . Proof:
The map
j : X* 5, Y*-H*
of 2.5 in section 2 is continuous
linear with I l j I ( = 1. Since H $ill Haydon’s result [21,Prop.3.1] tells us that B * = norm-clco (extBH*). We conclude, using observation4.1.3, H that BH* cnorm-clco (extBX*@extB * ) c norm-cl (j( B * Y x aT Y * ) ) cBH*, so that BH* = norm - cl (j(B * - y*)). Banach’s classical homomorphism ~~
theorem now reveals that j is a quotient map.
ExarnpZe: The James Tree space JT turned out towbe an example for a Banach space Z not containing 2, but such that Z B E Z does contain l l . Yet, according to Theorem 3.3.1 in section 3, JTZ,JT ought to be very close to not contain 21. In this sense, the James Tree space now also marks the limits of Theorem 4.1.6: The map
j : JT*S,,JT*-(JT&
4.2 THE EXTREME POINTS OF B
JT)* i s
not s u r j e c t i v e (1521).
(Kw* (X*,Y))*
The observation made in section 4.1 that extB(K *(x*,y))* is con-
tained in extB *@extBy* naturally leads to the queEtion whether there X is actually equality between the two sets, i.e. whether it is true that for x* and y* extreme points in BX+ and By*, respectively, the This problem functional x*@y* is an extreme point of B (K * (X*,Y))* ' had been solved in special cases by Singer r621 for C(K,X), Brosowski and Deutsch [ 3 1 and Strobele [671 for Co(S,X), S locally compact Haus-
dorff, and, for general completed injective tensor products X % E Y I by Hulanicki / Phelps 1311 and Tseitlin [691. C. Stegall and I [ 5 2 1 took up the general K (X*,Y)-case and proved the following result which W* shows that the extreme points in the dual unit ball of the operator space K ,(X*,Y) in fact are completely determined by the extreme points W in the duals of X and Y.
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W. Rums
4.2.1 Theorem ([52,Thm.1.1]): L e t H b e any Z i n e a r s u b s p a c e of T h e n e x t B H * = ( e x t BX * ) @ ( e x t B y * l . Kw,(X*,YI, containing X @ Y . 4.2.2 Corollary ([52,Thm.1.3]): L e t H be any l i n e a r subspace o f K(X, Yl, c o n t a i n i n g t h e f i n i t e - r a n k o p e r a t o r s . T h e n we h a v e : extBH, = (extB **lQ(extBy*l. X (For the special case of K(Z2), see Holub [30,Thm.3.1].) We established Theorem 4.2.1 in [52] by proving the following more general fact. Given X IBY c H c Kw* (X*,Y), h*EH* , Ilh*II=I , and Cx6,yZ) E extBX*XextBy*, such that h* I X B Y = 6 IX@Y , consider the set (x;5I Yb 1 M(BX*xBy*). Then we have:
(a) C is the set of norm-one extensions of h* to C(BX*xBy*), and (b) C is an extremal subset of BM(B *XB * ) . X Y Here, we give a direct proof of Theorem 4.2.1 based on Tseitlin’s result (thanks to S. Heinrich and W . Schachermayer). P r o o f of Theorem 4.2.1: Let ( x ~ , y ~ ) E e x t*xextBy*. B Then, according X to Tseitlin’s result, T = x*@y*IX%, Y E extB ( x s E y y * .Consider the set 0
0
(h* H* I h* IXgi,Y = T and Ilh*ll=I]. Then 0 is a weak*-compact convex extremal subset of BH*, and thus is the weak*-closed convex hull of its extreme points. Let htEext0. Then, since Q, is an extremal subset of BH*, h* is an extreme point of BH*, and thus, according to obser0 with x: and y: extreme vation 4.1.3 above, is of the form h* = x;@yT 0 points in B t and By*, respectively. Since h:0, we have: X = x:@y:IX@Y. This implies that x;@y: = x : % y : x;@yiIX@Y = h:IXOY is an extreme point of BH*. even on H. We conclude that 0 = ( x : @ y : ] @ =
5. GEOMETRY OF THE DUAL UNIT BALL OF Kw* (X*,Y) AND DIFFERENTIABILITY OF THE NORM After discovering that the extreme points of the dual unit ball of K *(X*,Y) are made up exactly by the set of functionals x * W * with x? and y* extreme points in the dual unit balls of X and Y, respectively, it was only natural to ask whether analogous results hold for the various special classes of extreme points. C. Stegall and I thus investigated the form of the [w*-]exposed and [w*-]strongly exposed points of the dual unit ball of Kw*(X*,Y). And, finally, Joe Diestel posed the problem of determining the denting points. We shall present in this section the respective results and discuss various applications. First, we briefly recall the definitions of these special classes of extreme points. For a detailed discussion and equivalent definitions,
Geometry of operator spaces
71
t h e reader is. r e f e r r e d t o J o e D i e s t e l ' s L e c t u r e N o t e s [ 9 1 a n d , concerning denting points, t o Phelps [ 4 4 ] . Definition:
L e t Z be a Banach s p a c e , and z o E Z w i t h I I z J l
=
1.
( a ) ( i ) z o i s a n e x p o s e d p o i n t of B Z i f t h e r e e x i s t s zzEZ*, I l z ~ l I = ,I s u c h t h a t z;(zo)=l and z* ( z ) < 1 f o r a l l zEBZ { z o ] . 0
The s e t of e x p o s e d p o i n t s of B Z w i l l b e d e n o t e d by e x p B Z .
(ii) I f Z = X* and t h e e x p o s i n g f u n c t i o n a l z*0 h a p p e n s t o b e a n e l e i s s a i d t o be a # * - e x p o s e d p o i n t of Bx* * 0 0 0 The s e t o f a l l s u c h p o i n t s w i l l b e d e n o t e d by w * - e x p B X t .
ment of X : z*=x EX, t h e n z (b)(i)
I1 z;lI=l
,
z
0
i s a s t r o n g l y e x p o s e d p o i n t of B
such t h a t
,
z;(zn) -1
2 ;
a n d , whenever
(zo)=l
t h e n llzn-zoll-O.
Z
i f t h e r e e x i s t s z:EZ*,
(zn ) n i n B Z i s s u c h t h a t
The s e t o f s t r o n g l y e x p o s e d p o i n t s of BZ w i l l b e d e n o t e d by s e x p B Z .
happens t o ( i i ) I f Z = X* and t h e s t r o n g l y e x p o s i n g f u n c t i o n a l b e an e l e m e n t of X : z * = x E X , t h e n zo i s s a i d t o b e a w * - s t r o n g l y 0
0
e x p o s e d p o i n t of BX*. The s e t of a l l s u c h p o i n t s w i l l b e d e n o t e d by w*-sexpB
*.
X ( c )( i ) z o i s a d e n t i n g p o i n t of BZ i f , g i v e n any E > 0 , t h e r e e x i s t a n d , whenever I l z l 1 1 1 , 6 > 0 and zZEZ*, I I z ~ l l = I , s u c h t h a t z;(z ) > 1 - 6 0 a n d z ; ( z ) > I - & , t h e n 1 I z - z 1 1 < E ( i . e . zo i s c o n t a i n e d i n s l i c e s o f 0
a r b i t r a r i l y s m a l l diameter).
The s e t of a l l d e n t i n g p o i n t s of B
Z ( i i ) I f Z = X* and t h e s l i c e s o f
b e g e n e r a t e d by e l e m e n t s z r
w*-denting
=
w i l l b e d e n o t e d by d e n t B Z .
( c ) ( i )c a n a l w a y s b e c h o s e n t o
x E i n X , t h e n z o i s s a i d t o be a
p o i n t o f EX*. The s e t of a l l s u c h p o i n t s w i l l b e d e n o t e d
w*-dentBX*.
by
5.1 Theorem 1 5 3 1 :
L e t H b e a c l o s e d l i n e a r s u b s p a c e of K w , ( X * , Y I ,
c o n t a i n i n g X % , Y . Then w e have:
(a)
[ w * - ] e s p BH * = [ w * - I e x p B X * @ [ w * - ] e x p BY *.
ibl
[w*-IsexpB H * = ~ w * - l s e x p B X ~ Q [ w * - l s e x p B ~ ~ .
(ci
[ w * - 1 d e n t B H * = [ w * - I d e n t B X * @ Iw * - l d e n t B y * . ( F o r t h e w*-part o f p r o p o s i t i o n
( a ) , compare J . A .
Johnson [ 3 2 1 . )
Thus, i n t h e v e i n of o u r g e n e r a l theme s e t a t t h e b e g i n n i n g of t h i s p a p e r , i n a l l cases, t h e geometry of t h e d u a l u n i t b a l l s o f X and Y c o m p l e t e l y d e t e r m i n e s t h e g e o m e t r y o f t h e , d u a l u n i t b a l l of t h e o p e r a t o r s p a c e K w J , ( X ,Y). A l t h o u g h t h e r e s u l t s of Theorem 5 . 1 q u a l i t a t i v e l y a r e a l l of t h e same n a t u r e , t h e r e s p e c t i v e methods of p r o o f t h e y r e q u i r e d t u r n e d o u t t o be q u i t e d i f f e r e n t from e a c h o t h e r and r a t h e r i n v o l v e d , s o t h a t it
72
W. Rues
would go beyond proof. Instead, general results We shall now
the scope of this survey to present even the ideas of the reader is referred to our paper L531, where more are derived. Some of these will be quoted below. discuss some consequences.
5.2 SOME EXTENSIONS AND CONSEQUENCES We first use Theorem 5.1 (b) to derive Feder / Saphar’s extension of Schatten’s result that, for H infinite-dimensional Hilbert, K ( H ) is not isometric to a dual space. 5.2.1 Theorem (Feder / Saphar [14,Thm.2]):
Assume t h a t X and Y
a r e r e f l e x i v e , and Z e t H b e any c l o s e d l i n e a r s u b s p a c e of K ( X , Y I , taining the finite-rank
operators.
I f H i s i s o m e t r i c t o a duaZ s p a c e ,
Proof:
con-
then H i s r e f l e x i v e .
Assume that H = Z*. According to our assumptions, Z** = H*
has RNP (Theorem 3.2.1 above).Hence, we have (taking into account that X and Y are reflexive): BZ** = norm-clco (sexpB * * ) . We now use z Theorem 5.1 (b) (and the fact that X and Y are reflexive) to conclude that BZ** = norm-clco (sexpBZ**) = norm-clco (sexpBH*) = = norm-cl co ( sexpBX**OsexpBy*) = = =
norm-clco (w*-sexpB **@w*-sexpBy*) = X norm-clco (w*-sexpBH*) = norm-clco (w*-sexpB * * ) =
z
norm-clco (sexpB ) = BZ . z (We used the fact that, for a general Banach space W, w*-sexpBW **= = sexpB . ) This shows that Z, and thus H, is reflexive. W =
To deduce further consequences, we first recall a classical result by Smul’yan connecting the exposed point structure of the dual unit ball with differentiability properties of the norm. Theorem ( Smul’yan [64,651 ) : L e t z o b e a n e l e m e n t of norm o n e of a Banach s p a c e 2. T h e n t h e norm o f 2 i s G a t e a u x - ( r e s p . F r s c h e t - l d i f f e r e n t i a b l e a t z o w i t h d i f f e r e n t i a l z,* i f a n d o n l y i f z o w * - e x p o s e s ( r e s p . w * - s t r o n g l y e x p o s e s ) BZ* a t z *
.
In our paper [531, instead of Theorem 5.1 (b), we prove a more general result which allows us to extend the assertion of this proposition to any linear subspace H even of Lw* (X*,Y). In particular, we can deduce the following result on the geometry of the dual unit balls of K(X,Y) and L(X,Y). 5.2.2 Corollary [531: L e t X*
Q
Y
t
H c L ( X , Y).
.
T h e n w*-sexpBH* = s e x P B X B w * - s e x p B y * I n p a r t i c u l a r , we h a v e : w * - ~ e x p B ~ ( ~ ,=~ w*) * s e x p B W ( X y,* , = w*-sexpB (X, y,* = s e x p B - * - s e x p B y * .
X
73
Geometry of operator spaces
Together with a result on G-differentiability that, again, is stronger than Theorem 5.1 (a), the extension of Theorem 5.1 (b) allows us to derive the following consequences. 5.2.3 Corollary [ 5 3 1 :
w i t h llz II=2=IlyolI.
L e t zo€X, yoEY,
( a ) Let XOYcHc3(X*,Y*l = LiX*,Y**). Then I I I I H i s F - d i f f e r e n t i a b z e a t zo@y if and o n l y i f II I1 IIY a r e F - d i f f e r e n t i a b l e a t xo and yo, r e s p e c t i v e l y . ( b ) Let -
XQY cHcCodd(BX*XBy*)
f(z*,-y*l
Then I1
I1 I I Y (
lb
= -j-(z*,y*)
(
f E C o d d ( B X * X B Y *):
IIx
$(-x*,y*)
and
=
).
is G - d i f f e r e n t i a b l e a t z Oy 0
ure G-differentiable
0
if and o n l y i f II I I x
and
a t xo and yo, r e s p e c t i v e l y .
For proposition (b), compare J.A. Johnson [321.
)
The extensions of Theorem 5.1 (a) and (b) can also be used to derive the results on F- and G-differentiability of the norms in K(X,Y) and L(X,Y) announced by S . Heinrich in [23]. We note here one particular case. L e t k o € K ( X , Y l w i t h IlkolI = 2 . T h e n 5.2.4 Corollary ( Heinrich ) : t h e following propositions are equivalent:
( a ) T h e norm of K ( X , Y I
is F - d i f f e r e n t i a b l e a t k o .
( b ) T h e norm of L ( X , Y )
is F-differentiable a t k o .
( c ) T h e norm of L ( X * * , Y * * )
i s F-differentiable
a t kz*.
(Actually, Heinrich [23,Cor.4.2] proved this up to the space
L(X,Y), i.e. the equivalence of (a) and (b).)
Some of the above results are significantly different from those of the preceding sections in that they transfer properties of X* and (X*,Y) but to the duals of much larger Y* not only to the dual of spaces of merely continuous inear operators. Further results in this direction can be found in the joint paper [ 5 3 1 with C. Stegall.
6. PROBLEMS
We close this paper with some of the problems that naturally arise from our discussion of the operator spaces K ( X , Y ) and Q,(X*,Y) in the previous sections. 6.1
When is it true that K(X,Y) does not contain (an isomorph of) co ?
6.2
Under which conditions does K(X,Y) have RNP ?
6.3
Is there an isomorphic version of Feder / Saphar’s result 5.2.1 above, i.e. if, for reflexive Banach spaces X and Y, the space K(X,Y) is isomorphic t o a dual space, is K(X,Y) then necessarily reflexive ?
74
W. Rues
6.4
Instead of %* (X*,Y) - and thus X Z E Y-and K(X,Y) - consider the completed projective tensor product X e T Y of Banach spaces X and Y: to what extent can the geometric and topological properties of X&Y and its dual L(X,Y*) be recovered from the corresponding properties of the factor spaces X and Y and their duals ?
6.5
To what extent can geometric and topological properties of particular operator spaces other than K(X,Y) - like p-absolutely summing, p-integral, or p-nuclear operators - be recovered from those of the factor spaces X and Y and their duals ? COMMENTS AND RELATED PROBLEMS
-
6.1 and 6.2: The general feeling is that like for reflexivity itself - for positive results on these weakenings of reflexivity for K(X,Y), the coincidence of L(X,Y) with K(X,Y) will play an important role. For the Radon-Nikodym property, this is being discussed in Diestel / Morrison [81. They showed: Theorem ( Diestel / Morrison [8] ) : S u p p o s e t h a t X* i s s e p a r a b z e o r r e f l e x i v e , a n d t h a t Y has R N P . T h e n , w h e n e v e r L ( X , Y I = K ( X , Y / ,
K (X,Y l h a s RNP. For extensions of this result, see Andrews [ l ] . Compare also the discussion of the RNP for operator spaces in Diestel / Uhl [10,Ch.VIII,p.258 For the problem of containing col results by A.E. Tong [681 and N.J. Kalton [351 indicate a strong connection with the coincidence of bounded and compact linear operators. Theorem ( Kalton [ 351 ) : S u p p o s e t h a t X has a n u n c o n d i t i o n a l f i n i t e - d i m e n s i o n a l e x p a n s i o n of t h e i d e n t i t y . T h e n , i f Y i s a n y i n f i n i t e - d i m e n s i o n a l Banach s p a c e , t h e f o l l o w i n g a r e e q u i v a l e n t : (a)
K(X,Yl $ e o
(el
K(X,YI
.
(bl
L(X,YI
i s complemented in L(X,Yl
.
= K(X,Y/
.
Thus, problems 6.1 and 6.2 bring our attention back to the "old" problem of characterizing spaces X and Y for which there exist non-compact bounded linear operators from X into Y, and to the problem, whether K(X,Y) ever is complemented in L(X,Y) in a "non-trivial" way : is it true that either L(X,Y) = K(X,Y), or K(X,Y) is uncomplemented in L(X,Y) ? For a thorough recent discussion of these latter problems, the reader is referred to J. Johnson [331.
6.3:
Problem 6.3 has earlier been raised by J.R. Retherford [48,p.10061. It leads us to the general problem of when K(X,Y) is (isometric or isomorphic to) a dual. In particular: is K(X) ever a dual ? J. Hennefeld [27,Cor.2.3] showed: I f X h a s a c o m p l e m e n t e d s u b s p a e e w i t h a n u n c o n d i t i o n a l b a s i s , t h e n K t X l is n o t i s o m o r p h i c t o a dual. And J. Johnson [33,Prop.l] proved the following result on the non-conjugacy of K(X,Y) :
75
Geometry of operator spaces I f Y has t h e bounded a p p r o x i m a t i o n p r o p e r t y ,
m e n t e d i n L(X,Y),
and K(X,Y)
i s n o t comple-
t h e n KIX,Y) i s n o t i s o m o r p h i c t o a c o m p l e m e n t e d sub-
s p a c e of a d u a l s p a c e .
Note that, again, an assumption on the relative position of K(X,Y) in L(X,Y) interferes ! For further information on the problem of non-conjugacy of K(X,Y) or L(X,Y), the reader may consult Hennefeld 1271 and J. Johnson [331. 6.4: This program particularly includes the problem of characterizing conditions on X and Y such that X%,Y has RNP, does not contain an isomorph of 11 or col or is weakly sequentially complete.
Note that this program is2uite ambitious, for it includes the case of the space L1 ( p , X ) = L1 (11) Bn X, which, just for the special case of weak sequential completeness and of characterizing weak compactness, turned out to be rather difficult. ( As far as I know, M. Talagrand finally showed that L1(p,X) is weakly sequentially complete whenever X is. ) A further particular problem is to find out under which conditions the space N(X,Y) of nuclear operators has RNP; compare the remarks following Corollary 3.2.2 in section 3 above.
.
In 1241, Heinrich showed that sexpBXgny = sexpB @sexpBy We deX rive this result in [53] as a further special case of our extended version of Theorem 5.1 in section 5 above.
6.5:
For this program, we only give a partial list of references related to this problem. Reflexivity of the space of p-absolutely summing operators (Ilp<-) is being discussed in Gordon/Lewis/Retherford 1171, and in Saphar 1561.
D.R. Lewis [38] discussed reflexivity, the RNP, and weak sequential completeness for various classes of a-tensor products, and specializations to corresponding classes of absolutely summing and integral operators. Heinrich [25] obtained conditions under which the spaces of p-integral and of p-absolutely summing operators (a) have RNP, and (b) do not contain (an isomorph of) c o . Non-containment of lI and of cOr and the RNP and weak sequential completeness for the spaces of p-nuclear and quasi-p-nuclear operators are being discussed in Makarov/Samarskii 1411. Andrews [I], too, has results on the RNP for the spaces of p-absolutely summing and of p-nuclear operators.
Finally, Heinrich [26] als_o obtained -results on the weak sequential Y [551 and of the spaces of completeness of the spaces X@gpY and X m
EP
p-absolutely summing and of p-nuclear operators. REFERENCES 1. 2.
ANDREWS,K.T.: T h e Radon-Nikodym p r o p e r t y f o r s p a c e s of o p e r a t o r s . J.London Math.Soc.(2)28 (1983), 113-122. BOURGAIN,J., TALAGRAND,M.: C o m p a c i t g e x t r s m a l e . Proc.Amer.Math. Soc.80 (1980), 68-72.
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BROSOWSKI,B., DEUTSCH,F.: On some g e o m e t r i c p r o p e r t i e s o f s u n s . J.Approx.Theory 10 (1974), 245-267. 4. COLLINS,H.S., RUESS,W.: Weak c o m p a c t n e s s in s p a c e s o f c o m p a c t o p e r a t o r s and of v e c t o r - v a l u e d f u n c t i o n s . Pacific J.Math.106 (1983), 45-71. 5. COLLINS,H.S., RUESS,W.: D u a l s o f s p a c e s o f c o m p a c t o p e r a t o r s . Studia Math.74 (1982), 213-245. 6. DE WILDE,M.: P o i n t w i s e c o m p a c t n e s s i n s p a c e s of f u n c t i o n s and R . C . J a m e s ' T h e o r e m . Math.Ann.208 (1974), 33-47. 7. DIESTEL,J.: T h e Radon-Nikodym p r o p e r t y and t h e c o i n c i d e n c e o f i n t e g r a l and n u c l e a r o p e r a t o r s . Rev.Roum.Math.Pures et Appl.17 (1972), 1611-1620. 8. DIESTEL,J., MORRISON,J.T.: T h e Radon-Nikodym p r o p e r t y f o r t h e s p a c e o f o p e r a t o r s . Math.Nachr.92 (1979)I 7-12. 9. DIESTEL,J.: G e o m e t r y of Banach s p a c e s - S e l e c t e d t o p i c s . Lecture Notes in Math., Vo1.485. Berlin, Hdbg.,New York. Springer 1975. 10. DIESTEL,J., UHL,J.J.Jr.: V e c t o r m e a s u r e s . Amer.Math.Soc.Math.Surveys 15 (1977). 1 1 . DUNFORDIN., SCHATTEN,R.: On t h e a s s o c i a t e and c o n j u g a t e s p a c e f o r t h e d i r e c t p r o d u c t . Trans.Amer.Math.Soc.59 (1946), 430-436. 12. DUNFORDIN., SCHWARTZ,J.T.: L i n e a r o p e r a t o r s I. New York, London. Interscience 1958. 13. FAKHOUR1,H.: Q u e l q u e s p r o p r i d t d s de l ' e s p a c e d e s o p d r a t e u r s comp a c t s . Studia Math.64 (1979), 239-247. 14. FEDER,M., SAPHARIP.: S p a c e s o f c o m p a c t o p e r a t o r s and t h e i r d u a l s p a c e s . Israel J.Math.21 (1975)I 38-49. 15. FLORET,K.: W e a k l y c o m p a c t s e t s . Lecture Notes in Math., V01.801. Berlin, Hdbg., New York. Springer 1980. 16. GIL DE LAMADRID,J.: M e a s u r e s and t e n s o r s . Trans.Amer.Math.Soc.114 (1965), 98-121. 17. GORDON,Y., LEWIS,D.R., RETHERFORD,J.R.: Banach i d e a l s of o p e r a t o r s w i t h a p p l i c a t i o n s . J.Functiona1 Analysis 14 (1973), 85-129. 18. GRAVES,W.H., RUESS,W.: C o m p a c t n e s s and weak c o m p a c t n e s s in s p a c e s o f c o m p a c t r a n g e v e c t o r m e a s u r e s . To appear in Can.J.Math. 19. GROTHENDIECK,A.: P r o d u i t s t e n s o r i e l s t o p o Z o g i q u e s e t e s p a c e s n u c l B a i r e a Amer.Math.Soc.Memoirs 16 (1955). 20. HAGLER,J.: Some more Banach s p a c e s w h i c h c o n t a i n 21. Studia Math. 46 (1973), 35-42. 21. HAYDON,R.: Some more c h a r a c t e r i z a t i o n s o f Banach s p a c e s c o n t a i n i n g 7-1. Math.Proc.Cambridge Phil.Soc.80 (1976), 269-276. 22. HEINRICH,S.: On t h e r e f l e x i v i t y o f t h e Banach s p a c e L ( E , F ) . Funct. Analysis Appl.8 (1974), 186-187. Translated from Funktional’nyi Analiz i Ego Prilozheniya 8 (1974). 23. HEINRICH,S.: The d i f f e r e n t i a b i l i t y of t h e norm i n s p a c e s o f o p e r a t o r s . Funct.Analysis Appl. 9 (1975), 360-362. Translated from Funktional’nyi Analiz i Ego Prilozheniya 9 (1975). 24. HEINRICHIS.: CTpOrO BHCTaBJIeHHhle II KOHHaeCKIIe TOgKU B IIpOeKTUBHOM TeH3OpHOM llpOH3Be,QeHUU. Teopux @yHKLlUU, @yHKU.AHaJ’M3 II HX nPUJIOKeHHR (XapbKOB) 22 (1975)I 146-154. 25. HEINRICHIS.: Some p r o p e r t i e s o f o p e r a t o r s p a c e s . Serdica Bulg. Math.Publ.3 (1977)I 168-175. 26. HEINRICH,S.: Weak s e q u e n t i a l c o m p l e t e n e s s o f Banach o p e r a t o r i d e a l s . Sib.Math.J.17 (1976), 857-862. Translated from Sibirskii Mat.Zhurna1 17 (1976). 27. HENNEFELD,J.: T h e n o n - c o n j u g a c y of c e r t a i n a l g e b r a s of o p e r a t o r s . Pacific J.Math.43 (1972), 111-113. 2 8 . HOLUB,J.R.: H i l b e r t i a n o p e r a t o r s and r e f l e x i v e t e n s o r p r o d u c t s . Pacific J.Math.36 (1971), 185-194. 29. HOLUB,J.R.: R e f l e x i v i t y of L f E , F ) . Proc.Amer.Math.Soc.39 (1973)I 175-1 77.
Geometry of operator spaces
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30. HOLUB,J.R.: On t h e m e t r i c g e o m e t r y of i d e a l s o f o p e r a t o r s on H i l b e r t s p a c e . Math.Ann.201 (1973), 157-163. 31. HULANICKI,A., PHELPS,R.R.: Some a p p l i c a t i o n s of t e n s o r p r o d u c t s o f p a r t i a l l y o r d e r e d s p a c e s . J.Funct.Analysis 2 (1968),177-201. 32. JOHNSON,J.A.: E x t r e m e p o i n t s i n t e n s o r p r o d u c t s and a t h e o r e m o f De Leeuw. Studia Math.37 (1971), 159-162. 33. JOHNSON,J.: Remarks o n Banach s p a c e s o f c o m p a c t o p e r a t o r s . J.Funct. Analysis 32 (1979), 304-311. 34. JUN,K.K.: A r e m a r k on a t h e o r e m o f R . C . J a m e s . Bol.Soc.Mat.Mexicana 16 (1971), 29-31. 35. KALTON,N.J.: S p a c e s of c o m p a c t o p e r a t o r s . Math.Ann.208 (19741, 267-278. 36. KHURANA,S.S. : P o i n t w i s e c o m p a c t n e s s on e x t r e m e p o i n t s . Proc.Amer. Math.Soc.83 (1981), 347-348. 37. LEWIS,D.R.: C o n d i t i o n a l weak c o m p a c t n e s s i n c e r t a i n i n d u c t i v e t e n s o r p r o d u c t s . Math.Ann.201 (1973), 201-209. 38. LEWIS,D.R.: D u a l s of t e n s o r p r o d u c t s . Lecture Notes in Math., V01.604 (1978), 57-66. 39. LINDENSTRAUSS,J., STEGALL,C.: E x a m p l e s of s e p a r a b l e s p a c e s w h i c h d o n o t c o n t a i n 21 a n d whose d u a l s a r e n o n - s e p a r a b l e . Studia Math.54 (1975), 81-105. 40. LUST,F. : P r o d u i t s t e n s o r i e l s i n j e c t i f s d ' e s p a c e s f a i b l e m e n t s d q u e n t i e l l e m e n t c o m p l e t s . Colloq.Math.33 (1975), 289-290. 41. MAKAROV,B.M. , SAMARSKI1,V.G. : C e r t a i n p r o p e r t i e s i n h e r i t e d b y t h e s p a c e s o f p - k e r n e l and q u a s i - p - k e r n e l o p e r a t o r s . Funct.Ana1ysis Appl.15 (1981), 227-229. Translated from Funktional’nyi Analiz i Ego Prilozheniya 15 (1981) 42. ODELL,E., ROSENTHAL,H.P.: A d o u b l e - d u a l c h a r a c t e r i z a t i o n of s e p a r a b l e Banach s p a c e s c o n t a i n i n g l ~ .Israel J.Math.20 (1975), 375-384. 43. PELCZYNSK1,A.: On Banach s p a c e s c o n t a i n i n g L1 ( p ) . Studia Math. 30 (1968), 231-246. 44. PHELPS,R.R.: D e n t a b i l i t y and e x t r e m e p o i n t s i n Banach s p a c e s . J.Funct.Analysis 16 (1974), 78-90. 45. PITT,H.R.: A n o t e on b i l i n e a r forms. J-London Math.Soc.11 (19361, 174-1 80. 46. PROLLA,J.B.: A p p r o x i m a t i o n of v e c t o r VaZued f u n c t i o n s . NorthHolland Math.Studies 25. Amsterdam, New York, Oxford 1977. 47. RAINWATER,J.: Weak c o n v e r g e n c e of b o u n d e d s e q u e n c e s . Proc.Amer. Math.Soc.14 (1963), 999. 48. RETHERFORD,J.R.: A p p l i c a t i o n s of Banach i d e a l s o f o p e r a t o r s . Bull. Amer.Math.Soc.81 (1975), 978-1012. 49. ROSENTHAL,H.P.: A c h a r a c t e r i z a t i o n of Banach s p a c e s c o n t a i n i n g 21. Proc.Nat.Acad.Sci.USA 71 (1974), 2411-2413. 50. RUCKLE,W. H. : R e f Z e x i v i t y o f L (E,F ) Proc .Amer .Math.SOC. 34 (1972), 171-174. 51. RUESS,W.: T h e s p a c e of c o m p a c t r a n g e v e c t o r m e a s u r e s . Proc.Conf. "Vector Measures and Integral Representations of Operators", Essen 1982. To appear in Vorl.Fachb.Math.Essen, Vol.10. 52. RUESS,W., STEGALL,C.: E x t r e m e p o i n t s i n d u a l s of o p e r a t o r s p a c e s . Math.Ann.261 (1982), 535-546. 53. RUESS,W., STEGALL,C.: E x p o s e d p o i n t s i n d u a l s of o p e r a t o r s p a c e s . In preparation. 54. SAMUEL,C.: R e p r o d u c t i b i Z i t 2 de II d a n s l e s p r o d u i t s t e n s o r i e l s . Math.Scand.50 (1982), 248-250. 55. SAPHAR,P.: T e n s o r p r o d u c t s o f Banach s p a c e s and c l a s s e s of l i n e a r m a p p i n g s . Studia Math.38 (1970), 71-100. 56. SAPHARIP.: H y p o t h e s i s of a p p r o x i m a t i o n of o r d e r p i n Eanach s p a c e s and a p p r o x i m a t i o n o f p - a b s o l u t e l y summing m a p p i n g s , Israel J. Math.13 (19721, 379-399.
.
.
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57. SCHACHERMAYER,W. : S u r un thbore'me de G r o t h e n d i e c k . Sgminaire P. Lelong 1974/75. Lecture Notes in Math., Vo1.524. Berlin, Hdbg., New York. Springer 1976. 58. SCHATTEN,R.: A t h e o r y o f c r o s s s p a c e s . Annals of Math.Studies 26. Princeton, N.J. Princeton Univ.Press 1950. 59. SCHATTEN,R.: The s p a c e o f c o m p l e t e l y c o n t i n u o u s o p e r a t o r s on a H i l b e r t s p a c e . Math.Ann.134 (1957), 47-49. 60. SCHWARTZ,L.: T h b o r i e d e s d i s t r i b u t i o n s a' v a l e u r s v e c t o r i e l l e s I . Ann.Inst.Fourier (Grenoble) 7 (1957), 1-139. 61. SCHMETS’J.: S p a c e s of v e c t o r v a l u e d c o n t i n u o u s f u n c t i o n s . Lecture Notes in Math., Vo1.644, 368-377. Springer 1978. 62. SINGER,I.: S u r l a m e i l l e u r e a p p r o x i m a t i o n d e s f o n c t i o n s a b s t r a i t e s c o n t i n u e s a' v a l e u r s d a n s un e s p a c e de B a n a c h . Rev.Roum.Math. Pures Appl. 2 (1957), 245-262. 63. SINGER,I.: S U P l e s a p p z i c a t i a n s Z i n k a i r e s i n t d g r a l e s d e s e s p a c e s de f o n c t i o n s c o n t i n u e s I . Rev.Roum.Math.Pures Appl.4 (1959), 391-401. 64. SMUL’YAN,V.L.: O n some g e o m e t r i c a l p r o p e r t i e s of t h e u n i t s p h e r e i n t h e s p a c e o f t h e t y p e ( B ) . Math.Sbornik N.S.6(48) (1939), 77-94. 65. SMUL’YAT3,V.L.: S U P l a d k r i v a b i l i t b de l a norme d a n s I ' e s p a c e de Banach. Doklady Akad.Nauk.SSSR (N.S.) 27 (1940), 643-648. 66. STEGALL,C.: T h e Radon-Nikodym p r o p e r t y in c o n j u g a t e Banach s p a c e s . Trans.Amer.Math.Soc.206 (1975), 213-223. 6 7 . STRdBELE,W.J.: O n t h e r e p r e s e n t a t i o n of t h e e x t r e m a l f u n c t i o n a l s on C o ( T , X ) . J.Approx.Theory 10 (1974), 64-68. 68. TONG,A.E.: On t h e e x i s t e n c e of n o n - c o m p a c t b o u n d e d l i n e a r o p e r a t o r s b e t w e e n c e r t a i n Banach s p a c e s . Israel J.Math.10 (1971), 451-456. 69. TSEITLIN,I.I.: T h e e x t r e m e p o i n t s o f t h e u n i t b a l l s of c e r t a i n s p a c e s o f o p e r a t o r s . Mat.Zametki 20 (1976), 521-527. 70. TSITSAS,L.N.: On weak s e q u e n t i a l c o m p l e t e n e s s i n b i p r o j e c t i v e t e n s o r p r o d u c t s p a c e s . Proc.Amer.Math.Soc.60 (1976), 119-123. 71. TSITSAS,L.N.: O n weak c o m p a c t n e s s i n b i p r o j e c t i v e t e n s o r p r o d u c t s p a c e s . Math.Nachr.78 (1977), 297-308. 72. TSITSAS,L.N.: On t h e s e m i - r e f l e x i v i t y o f b i p r o j e c t i v e t e n s o r p r o d u c t s p a c e s . J.Math.Analysis App1.69 (1979), 505-510. 73. TWEDDLE,I.: T o p o l o g i e s o f p o i n t w i s e c o n v e r g e n c e on f a m i l i e s o f e x t r e m a l p o i n t s and weak c o m p a c t n e s s . Proc.Amer.Math.Soc.86 (1982)I 85-86.
DUALITY AND GEOMETRY OF SPACES OF COMPACT OPERATORS Wolfgang Ruess * ) Fachbereich Mathematik Universitat Essen Universitatsstr.3, D-43 Essen Federal Republic of Germany
INTRODUCTION
The object of this paper is to discuss the problem of how - for given Banach spaces X and Y - the space K(X,Y) of compact linear operators from X into Y and its dual space reflect the geometric and topological properties of X and Y and their respective duals. Since X* and Y are closed linear subspaces of K(X,Y), they inherit trivially properties such as non-containment of Z 1 , the dual having the Radon-Nikodym property, weak sequential completeness, or reflexivity, from K(X,Y). The natural question thus is to find out which additional conditions are needed for the reverse implications: how can geometric and topological properties of K(X,Y) and its dual be recovered from the corresponding properties of (the presumably well known spaces) X and Y and their duals? Besides the properties already mentioned, we shall consider the representation of the dual of K(X,Y), and discuss how the various classes of extreme points of the dual unit ball of K(X,Y) can be represented by the corresponding classes of points in the (bi-) duals of X and Y. The paper is essentially written in survey form, but, except for section 5, the main stream results are presented with proofs. The spectrum of results considered here will cover only a very small sector of the subject matter and is restricted to those parts that I have been involved with during the past few years. PRELIMINARIES 1.1 SPACES OF COMPACT OPERATORS
1.
Our discussion is placed in the general context of the operator space Kw*(X*,Y) of compact and weak*-weakly continuous linear operators from X* into Y, endowed with the usual operator norm. This space, originally introduced by L. Schwartz [ 6 0 ] in 1957 as the so-called &-product XEY of X and Y , apparently h a s gone out of sight over the years. However, it has the advantage over K(X,Y) that, as far as methods of proofs are concerned, it is conceptually easier to handle than K(X,Y) itself, and that it comprises not only spa_es of the type K(X,Y) but also completed injective tensor products X O E Y , and thus many more concrete spaces of analysis particularly spaces of vectorvalued continuous functions and of vector-valued measures. More specifically, we have the following (well known) fundamental isometrical isomorphisms and isometrical embeddings, respectively:
-
This paper is based on joint work with H . S . Collins, Baton Rouge (Louisiana) [4,51 and C.P. Stegall, Linz (Osterreich) [52,531.
*)
K(X,Y)
k
-
W. R i m s
= Kw,(X**,Y)
k**
X % , Y U K w * (X* , Y )
x8y-
{x*-(x*x)yl,
-
and
X2,Y
= K
W*
X o r Y has t h e approximation p r o p e r t y , c f .
(1.2.a)
C(K,X)
F
(1.2.b)
= K,,(x*,c(K))
[601.
= C ( K ) % ~ X
{ x * h x*F)
c c a ( C , X ) = Kw,(X*,ca(C)) = c a ( 1 ) S 0
(X*,Y) i f
( K compact Hausdorf f )
EX
X*@)
{X*
(Here, C i s a u - a l g e b r a (of s u b s e t s of a nonempty s e t Q ) , and c c a ( C , X ) d e n o t e s t h e s p a c e o f c o u n t a b l y a d d i t i v e m e a s u r e s from C i n t o X w i t h r e l a t i v e l y compact r a n g e , endowed w i t h t h e s e m i - v a r i a t i o n norm, c f .[18,51] .) D u a l i t y r e s u l t s f o r Kw,(X*,Y) t h L s o f t e n a l l o w immediate s p e c i a l i z a t i o n s t o b o t h K ( X , Y ) - s p a c e s and X@,Y-spaces, and t h i s i s t h e g e n e r a l approach t h a t w e u s e i n t h i s p a p e r . 1 . 2 NOTATION
B a s i c a l l y , w e f o l l o w t h e c l a s s i c a l n o t a t i o n of Dunford and Schwartz [ 1 2 ] . A s u b s e t A o f a Banach s p a c e X i s c a l l e d v e a k l y s e q u e n t i a l l y p r e c o m p a c t i f e v e r y s e q u e n c e i n A h a s a weak Cauchy s u b s e q u e n c e . The u n i t b a l l of a Banach s p a c e X i s d e n o t e d by BX, and i t s e x t r e m e p o i n t s by e x t B X . The s p a c e s of a l l bounded, weakly compact, c o m p a c t , and n u c l e a r o p e r a t o r s from X i n t o Y - X and Y Banach s p a c e s - a r e d e n o t e d by L(X,Y) , W ( X , Y ) , K ( X , Y ) , * a n d N ( X , Y ) , r e s p e c t i v e l y . I n a c c o r d a n c e w i t h o u r d e f i n i t i o n of Kw,(X , Y ) , w e d e n o t e by L w , ( X z , Y ) t h e s p a c e of weak*-weakly c o n t i n u o u s l i n e a r o p e r a t o r s from X i n t o Y . The t e r m Radon-Nikodym p r o p e r t y i s , a s u s u a l , a b b r e v i a t e d by R N P , and t h e t e r m [ m e t r i c ] a p p r o x i m a t i o n p r o p e r t y by [ m . ] a . p .
2 . EARLY RESULTS W e b r i e f l y r e c a l l s e v e r a l c l a s s i c a l r e s u l t s on t h e d u a l i t y and geometry of K ( X , Y ) i n o r d e r t o g i v e a f l a v o u r of t h e r e s u l t s t o b e
expected 2.1
-
and t h o s e n o t t o be e x p e c t e d .
Dunford
/ S c h a t t e n [ I l l , 1 9 4 6 , and S c h a t t e n [ 5 8 1 , 1950:
I c o n t a i n s a c o p y of co i s o r n e t r i c a Z 2 y . P 2 . 2 S c h a t t e n [ 5 9 ] , 1957: The u n i t b a l l of KI121 has no e x t r e m e
For 1 < p < m
, t h e space
X ( l
points. 2 . 3 C o r o l l a r y ( Dunford / S c h a t t e n ) : L e t H be an i n f i n i t e - d i m e n s i o n a l H i Z b e r t s p a c e . T h e n K ( H 1 i s n o t r e f l e x i v e , and n o t e v e n w e a k l y s e q u e n t i a l l y c o m p l e t e , n o r i s o m e t r i c t o a dual. s p a c e .
These a r e , i n a s e n s e , t h e n e g a t i v e r e s u l t s . The message i s t h a t o n e c a n n o t e x p e c t K ( X , Y ) t o b e " n i c e " p r o v i d e d t h a t o n l y X and Y are " n i c e " enough. On t h e c o n t r a r y , K ( X , Y ) may j u s t become t o o l a r g e .
61
Geometry o f operator spaces
In the other direction, there are the following classical positive results. 2.4 Pitt t451, 1936:
For 1 I p < q < m , a l L b o u n d e d l i n e a r o p e r a t o r s
L (lq, Zpi = Kilq, Lpl . Combined with Grothendieck’s result 2.6(b)(i) quoted below, this implies that f o r 1 $ p < g < - , t h e s p a c e K ( Z l i i s r e f l e z i v e . q’ P
from L
9
into l
are compact:
2.5 Grothendieck [19], 1955: j
:
x*%,Y* x*-*
-
C o n s i d e r t h e c a n o n i c a l ( L i n e a r ) map
(K,,(x*,Y))*
(h+(hx*,y*)}.
T h e n we h a v e :
is s u r j e c t i v e i f X* o r Y* has R N P . ( b ) j i s i n j e c t i v e if X* o r Y* h a s a . p . More s p e c i f i c a L l y : I f e i t h e r o f X* and Y * h a s a . p . , (a)
j
p : X* g,Y* x*w*
-
i s i n j e c t i v e , and if p i s i n j e c t i v e ,
B(X,Y)
z(x,Y)-
then the map
(x*x)(Y*Y)I
t h e n s o is j .
2.6 Corollary (Grothendieck [ 1 9 ] , 1955): ( a ) I f X** o r Y * h a s R N P , t h e n K(X,Y)* is a q u o t i e n t of X * * & Y * , and if, i n a d d i t i o n , e i t h e r of X * * and Y * h a s a . p . , t h e n we h a v e : K I X , Y I * = X**-@,Y* = N ( X * , Y * ) . ( b l Assume t h a t X and Y a r e r e f l e x i v e . T h e n we h a v e :
( i ) If L(X,Y) = K(X,Y/, t h e n K(X,YI
i s reflexive.
( i i l C o n v e r s e l y , if K(X,Y/ is r e f l e x i v e and p : X**ZnY*-
is i n j e c t i v e ,
then LIX,Yi
B(X*,Y)
= K(X,Y/.
We pause for a proof of Corollary 2.6, for these consequences of Grothendieck’s results 2.5 later on have often been reconsidered (see Remarks 2.7 below)
.
P r o o f of Corollary 2.6: Proposition (a) is - via the isometry ( 1 . 1 ) just a special case of 2.5. For a proof of proposition (b), assume now that X and Y are reflexive. Then, according to (a), we have A quick inspection K(X,Y)* = XSTY*/Q, so that K(X,Y)** = Q’cL(X,Y). of the corresponding isometrical embeddings reveals that K(X,Y) is reflexive in case L(X,Y) = K(X,Y). This proves proposition (b)(i). As for (b)(ii), we need only use 2.5 to conclude from K(X,Y)*=XG, Y* (and the assumption) that K(X,Y)=K(X,Y)**=(XG, Y*)*=L(X,Y).
2.7 Remarks: 1. Theorem 2.5 is possibly better known as the result on the coincidence of integral and nuclear operators in the
62
W. R u e s
presence of RNP. Grothendieck [_?9,1.54.2,Thm.8]did not explicitly state 2.5, but proved it for XQ9,Y under the assumpion that X is reB(X,Y) is flexive or has a separable dual, and that p : X* any*injective. In his proof, however, he only uses that, under the assumptions on X, the dual unit ball BX* has what he called the "Phillips property" ("proprigtg @ " ) [ 19,1.54.1 ,Def.6,p.104] which, in present day language, exactly is the RNP for X*. For an explicit proof of 2.5, see Diestel [7], Gil de Lamadrid [161, Schachermayer [571, and C. Stegall [66,Cor.71. 2 . The results of Corollary 2.6, only implicit in Grothendieck’s thesis [I91 - as indicated above - have explicitly been proved by various authors. Feder and Saphar [14,Thm.l] proved proposition (a), also relying heavily on Grothendieck’s work [191. Proposition (b) appears in various (partly weaker) versions in Ruckle [501, Holub [28,291, Kunas K. Jun [341, Kalton r35,section 2,Cor.21, and, in the form stated above, in Heinrich 1221.
3. Note that proposition 2.6(b)(ii) particularly implies Dunford / Schatten’s result that, for an infinite-dimensional Hilbert space HI K(H) is not reflexive. Historically, looking back, it is not too surprising that after the early results on spaces of operators on Banach spaces mentioned so far, not much happened in this direction for about a decade, and that it took the vivid revival of Banach space theory in the sixties to renew the interest in the geometric and topological structure also of operator spaces. This process did not start before the early seventies. A survey on a small part of it is the subject of the subsequent sections.
Note: It seems worth specifying the assertion of Theorem 2.5 in the sense of our general program - to determine K(X,Y) and its dual from the spaces X and Y and their duals: (a) If either of p*and Y* has RNP, then every T E K(X,Y)* has a ZI , and representation of the form T = XX.x**@y* , with 1 1 1 XI*)^ and ( ~ f bounded ) ~ sequences in X** and Y*, respectively, in the sense that Tk = thi(k**xr*,yf) for all k E K(X,Y). (b) If X* has RNP, then every Q E C(K,X)* (K compact Hausdorff) has and a representation of the form Q = .EXixfOui , with (Ai)iE Z,, bounded sequences in X* and M (K), respectively, in (xf) and (pi) the sense that
Q F = tAi(xfFdpi
for all FEC(K,X).
This is to be noted as the advantage over the case for general XI where the dual of C(K,X) can only be represented as a space of more general X*-valued measures, cf. Singer [631, Prolla [46,Ch.V.51 and Schmets [61,p.368-3771.
63
Geometry of operator spaces
3. REFLEXIVITY AND WEAKER NOTIONS We first recall the implications among the various weakenings of reflexivity.
Z*
RNP
/
Z
reflexive
Z $
-
Z
weakly sequentially complete
Bounded subsets of Z are weakly sequentially precompact.
l1
Any of these properties is being inherited by X and Y from the operator space Kw,(X*,Y). According to our general theme, we now turn to the discussion of which additional assumptions are required for the reverse implications. 3.1 REFLEXIVITY AND WEAK SEQUENTIAL COMPLETENESS Generally speaking (and modulo the a.p.) , for K(X,Y) to be reflexive or just weakly sequentially complete, all bounded operators from X into Y must necessarily be compact. More specifically, the following results hold.
3.1.1
Grothendieck [I91 (see 2.5 and 2.6 above):
Assume t h a t X and Y a r e r e f l e x i v e and t h a t e i t h e r o f X and Y has a . p . T h e n K(X,Y) is r e f l e x i v e if and o n l y if L(X,YI = K I X , Y / .
3.1.2 Theorem ( D.R. Lewis [37,Thm.2.1]):
( a ) Assume t h a t X and Y a r e w e a k l y s e q u e n t i a l l y c o m p l e t e , and t h a t Lw,(X*,YI
= Kw,(X*,Y).
Then Kw,IX*,Y/
is w e a k l y s e q u e n t i a L l y c o m p l e t e .
( b ) C o n v e r s e l y , a s s u m e t h a t Kw* IX*, Y ) is w e a k l y s e q u e n t i a L l y c o m p l e t e ,
and t h a t e i t h e r of X and Y has t h e m e t r i c a p p r o x i m a t i o n p r o p e r t y . T h e n X and Y a r e w e a k l y s e q u e n t i a l l y c o m p l e t e , and L w * I X * , Y )
= Ku,(X*,YI.
N o t e s : Proposition (a) as stated here is to be-found in [4,Thm.3.41. D.R. Lewis [37,Thm.2.1] proved it for the space X&,Y and thus needed the a.p. in one of the factors also for this part of the assertion. A special case of proposition (a) is the result of F. Lust [401.
What is behind Theorem 3.1.2 is the following general result.
3.1.2’ [4,Prop.3.11: Kw,(X*,YI i s w e a k l y s e q u e n t i a l l y c o m p l e t e if and o n Z y if (i) x and Y a r e w e a k l y s e q u e n t i a l l y c o m p l e t e , and (ii) Kw* (X*, Y ) is w e a k - o p e r a t o r - t o p o l o g y s e q u e n t i a l l y c l o s e d in Lw* tX*, Y). Thus, D.R. Lewis’ particular achievement - with his beautiful proof of proposition 3.1.2(b) in [371 - was to show that condition (ii) in 3.1.2’ together with the m.a.p. in one of the factors implies equality of Lw*(X*,Y) and Kw* ( X * , Y ) .
W. Rues
64
3.1.3 Corollary (D.R. Lewis [37,Cor.2.41, and [4,Thm.3.51):
( a ) A s s u m e t h a t X* and Y a r e w e a k l y s e q u e n t i a l l y c o m p l e t e and t h a t W(X,Yl
= K(X,Y/.
Then K ( X , Y I
i s weakly sequentiaZZy complete.
( b ) ConverseZy, assume t h a t K f X , Y ) i s w e a k l y s e q u e n t i a l l y c o m p l e t e and t h a t e i t h e r o f X* and Y has m . a . p . q u e n t i a l l y c o m p l e t e and W ( X , Y I
T h e n X* and Y a r e w e a k l y s e -
= K(X,Y).
This is just a special case of 3.1.2 by using the isometries K(X,Y) = Kw,(X**,Y) and W(X,Y) = Lw,(X**,Y). For further special cases (and extensions to particular locally convex spaces), the reader is referred to section 3 of [41, and to Tsitsas [70,711. Note, finally, that 3.1.3 implies Dunford / Schatten’s result that K(H) is not weakly sequentially complete if H is an infinite-dimensional Hilbert space. 3.2 THE DUAL HAVING THE RADON-NIKODYM PROPERTY In contrast with the situation for reflexivity, weak sequential completeness and non-containment of Z1 (3.3 below) - where additional assumptions are needed - the RNP for both X* and Y* completely determines the RNP for the dual of Kw* (X*,Y)
.
3.2.1 Theorem [52,Thm.1.9]:
T h e d u a l o f Kw*
X*,YI
h a s R N P i f and
onZy i f X* a n d Y* h a v e R N P .
3.2.2 Corollary [52,Cor.1.101: The d u a l o f K X,Yl o n l y i f X**
has R N P i f and
and Y * h a v e R N P .
A particular consequence of Theorem 3.2.1 is the result that - in
case either of X* and Y* has a.p. - the space N(X,Y*) = X*%?Y* of nuclear operators from X into Y * has RNP, provided that X* and Y * have RNP; for, under these assumptions, we have Kw,(X*,Y) = X5,Y
and
Kw,(X*,Y)* = X*Gi,Y* = N(X,Y*) (see 2.5 above). This special case of Theorem 3.2.1 is proved in Diestel / Uhl [IO,VIII.4,Thm.7,p.249], and it is their method of proof that we generally follow in our proof of Theorem 3.2.1. P r o o f of Theorem 3.2.1 :
We use the fact that
Z*
has RNP if and
only if every separable subspace of Z has a separable dual: Uhl, cf. [10,111.3,Cor.6,p.82] and Steqall 1 6 6 1 . From this equivalence, the necessity part of Theorem 3.2.1 follows trivially. NOW, assume that X* and Y* have RNP, and let A be a closed separable subspace of Kw* (X*,Y) Choose a dense sequence (hn)n in A, and a closed separable subspace Yo of Y such that imhncYo for all nElN. Accordingly, for
.
(h:)n c Kw* ( (Yo)* ,X), choose a closed separable subspace Xo of X such Thus, we can view the sequence (hn)n as a that imh,*CXo for all nElN
.
65
Geometry o f operator spaces
*,Yo).N o w , it i s g e n e r a l l y t r u e ( c f .
s u b s e t o f Kw* ( (X,)
M and N c l o s e d l i n e a r s u b s p a c e s o f X a n d Y ,
[ 601) t h a t f o r
respectively,
t h e space
Kw* (M*,N) i s ( i s o m e t r i c a l l y ) a c l o s e d l i n e a r s u b s p a c e o f Kw* (X*,Y) T h i s r e v e a l s t h a t A i s a c l o s e d l i n e a r s u b s p a c e of Kw* ((Xo)* , Y o ) . (Xo)* and (Yo)* a r e s e p a r a b l e ,
a s s u m p t i o n on X and Y , b o t h
.
By
so t h a t
(Xo)* gT (Yo)* i s s e p a r a b l e as w e l l . A c c o r d i n g t o Theorem 2.5 i n s e c t i o n 2 , t h e dual of
%* ( ( X o ) * ,Yo)
i s s e p a r a b l e . Consequently,
i s a q u o t i e n t of
l1
A s s u m e t h a t X and Y do n o t c o n t a i n 2 1
and t h a t e i t h e r o f X* and Y * has R N P . T h e n K m , l X * , Y I t h e isometry K ( X , Y )
= Kw+(X**,Y) o f
t h e f o l l o w i n g s p e c i a l i z a t i o n t o K(X,Y) 3.3.2
Corollary [4,Cor.1.12]:
11, and t h a t X**
(Yo)* , a n d t h u s
This completes t h e proof.
O F ( AN ISOMORPH O F )
3 . 3 . 1 Theorem [ 4 , T h m . 1 . 1 4 ] :
A s usual,
ST
t h e d u a l of t h e c l o s e d l i n e a r s u b s p a c e A
o f Kw,((Xo)* ,Yo) i s s e p a r a b l e a s w e l l . 3 . 3 NON-CONTAINMENT
(Xo)*
o r Y* h a s R N P .
.
d o e s n o t c o n t a i n 21.
( 1 . 1 ) above a l l o w s
Assume t h a t X* and Y do n o t c o n t a i n Then K l X , Y )
does n o t c o n t a i n l l .
N o t e s : 1 . The r e s u l t o f Theorem 3.3.1 h a s b e e n p r e s e n t e d by Heron C o l l i n s a t t h e 1980 C o n f e r e n c e on "Measure T h e o r y a n d i t s A p p l i c a t i o n s " a t N o r t h e r n I l l i n o i s U n i v e r s i t y , DeKalb, I l l i n o i s , a n d f i r s t a p p e a r e d i n o u r j o i n t announcement o f r e s u l t s of o u r p a p e r [ 4 1 i n t h e P r o c e e d i n g s o f t h a t c o n f e r e n c e i n 1981, e d i t e d by G . A . G o l d i n a n d R . W h e e l e r , p.187-192. I t i s i n t e r e s t i n g t o n o t e t h a t B.M. Makarov and V . G . Samarsk i i announced t h e a s s e r t i o n of C o r o l l a r y 3 . 3 . 2 i n [ 4 1 , T h m . 2 . 3 ] , and t h a t , i n t h e p e r i o d when o u r p a p e r [ 4 1 w a s i n p r i n t , C . Sam-uel's paper 1541 a p p e a r e d where h e p r o v e s 3.3.1 f o r t h e s u b s p a c e X a C Y. T h i s d e v e l o p m e n t may h e l p t o s u p p o r t o u r p o i n t - s p e c i f i e d i n s e c t i o n 1 . 1 t h a t t h e " r i g h t " s p a c e t o b e l o o k e d a t i s t h e s p a c e Kw* (X*,Y)
.
2 . F a k h o u r i [13,Thm.3] is separable.
proved 3.3.2
P r o o f o f Theorem 3 . 3 . 1 :
u n d e r t h e a s s u m p t i o n t h a t X**
We u s e R o s e n t h a l ' s f u n d a m e n t a l r e s u l t
(see R o s e n t h a l [491 and Ode11 / R o s e n t h a l [ 4 2 ] ) t h a t a Banach s p a c e Z does n o t c o n t a i n
( a n i s o m o r p h ) o f Z,
i f and o n l y i f t h e bounded s u b -
s e t s of 2 are w e a k l y s e q u e n t i a l l y p r e c o m p a c t . Assume t h a t X + Z 1 Kw,(X*,Y) = K,,(Y*,X).)
a n d t h a t Y* h a s RNP. Let
( N o t i c e t h e symmetry ( h n ) n b e a bounded s e q u e n c e i n Kw,(X*,Y).
Then t h e r e e x i s t s a s e p a r a b l e
( c l o s e d l i n e a r ) s u b s p a c e Yo of Y s u c h
t h a t imhncYo f o r a l l n€lN. According t o o u r assumption, parable,
see S t e g a l l [ 6 6 1 .
Let
(YE),
(Yo)* i s se-
b e a dense sequence i n (Yo)*. Then (hGy;)n i s a bounded s e q u e n c e i n X a n d t h u s , s i n c e X 4 Z l , h a s
a weak Cauchy s u b s e q u e n c e
(h;tkyl;)k. C o n s i d e r i n g now t h e s e q u e n c e
66
W. Rues
(h$ky;)k
in X, continuing inductively, and using a diagonal process,
we arrive at a subsequence (hnl)l of (hn)n such that (hzlyz)l is weakly Cauchy in X for all kE7N. Taking adjoints, we conclude that (hnlx*) 1 * ) )-Cauchy in Yo for all x*EX*. But, since (yi), is normis a in (yk k* , on bounded subsets of Y dense (Yo) the weak topology o(Yo, (Yo)*) 0'
of Yo and the topology u (Yo, (yZlk) coincide. Hence, (hnlx*)l is weakly Cauchy in Yo, and thus in Y, for all x* € X*. At this point, we use the isometrical embedding
-
Kw* (X*,Y)
C (BX*XBy*)
h {(x*,y*)-(hx*,y*)I , to conclude from the classical weak convergence result for C(K)-spaces that (hnl)
.
is weakly Cauchy in Kw* (X*,Y) This completes the proof.
Going back to Theorem 3 . 3 . 1 ,
made heavy use of Y* having RNP
and inspecting its proof
-
-
where we
the question arises naturally whether
the assumption of the RNP for either of the duals is only a technical matter that could possibly be dispensed with by just requiring X and Y not to contain l I . Too much hope in this direction is put to rest by the following example. 3.3.3
Example [52]: L e t J T d e n o t e t h e J a m e s T r e e s p a c e , [ 3 9 ] , but J T Z ; , J T d o e s c o n t a i n Z 1 .
cf.
[39].
Then J T $ I I
Proof:
JT is the dual of a (separable) Banach space B such that
B**/B = 1; = Hilbert space of the dimension of the continuum [ 3 9 1 .
Using the fact that projective tensor products "respect" quotients,
we conclude that JT*Zn JT* has l ; ~ T e as l ~ a quotient. NOW, the "diagonal" in Z;z, l ; is isometric to l 1 , which thus can be "lifted" isomorphically to JT* 3, JT* , cf. Pelczynski [ 43 ,Lemma 3.1 1. But JT* JT* is a closed linear subspace of (JTGEJT)* (see 2.5 in sec-
sTI
tion 2 ) , so that the dual of the separable space JT& JT contains I f . Using Pelczynski’s and Hagler’s results [43,201, we conclude that JT~, JT= z l . A combination of Corollary 3.3.2 with Theorem 2 . 5 yields the following connection between reflexivity and weak sequential completeness of K(X,Y).
67
Geometry o f operator spaces
L e t X and Y be r e -
3.3.4 Theorem ([4,Thm.3.91 and [5,Cor.3.31):
fZexive,
and c o n s i d e r t h e f o l l o w i n g s t a t e m e n t s :
(a) LJX,Y) (el K(X,Y/
= K(X,Yl; f b ) L(X,Y) i s reflexive; i s r e f l e x i v e ; ( d ) KlX,Y) i s w e a k l y s e q u e n t i a l l y c o m p l e t e ;
T h e n ( a ) i m p l i e s ( b ) , ( b ) i m p l i e s ( e l , and (c) i s e q u i v a l e n t t o i d ) . Pinally,
i f t h e n a t u r a l map
j : x~~~Y*-(K(x,YJ)* i s i n j e c t i v e ,
t h e n ( d ) i m p l i e s (a). j is i n j e c t i v e , w h e n e v e r X o r Y h a s a . p .
For a description of general weakly sequentially precompact subsets of Kw* (X*,Y) via the corresponding concepts in the factor spaces X and Y, the reader is referred to Lewis [37,Thm.3.11, and to [4,section 1 1 .
4. WEAK COMPACTNESS IN K(X,Y), AND EXTREME POINTS IN THE DUAL OF K(XtY)
4.1 WEAK COMPACTNESS W e start with the well-known fact that weak sequential convergence and weak compactness are determined by convergence not necessarily on all elements of the dual but just on the extreme points of the dual unit ball.
4.1.1 Rainwater [47]:
A bounded sequence
Z c o n v e r g e s w e a k l y t o z E 2 i f and o n l y i f
( z n l n i n a Banach s p a c e
(z*z
f o r a l l z*EextBZ*.
)
n n
converges t o z*z
This result was the starting point for an intense study of the reduction of weak sequential convergence and weak compactness in Banach and more general locally convex - spaces Z to the corresponding notions with respect to the weak topology o(Z,extBZ*) on Z generated by just the extreme points of B *. W e refer the reader to K. Floret’s detailed exposition of this devefopment in [ 151. For our purpose here, it suffices to recall one of the latest results in this direction. 4.1.2 Bourgain / Talagrand [2]: A b o u n d e d s u b s e t H o f a Banach s p a c e Z is w e a k l y r e l a t i v e l y c o m p a c t i f and o n l y i f i t i s r e l a t i v e l y c o u n t a b l y compact w i t h r e s p e c t t o o ( Z , e x t B Z * l .
Note that, in result 4.1.2, the improvement over RainR e m a r k s : 1. water’s result - for describing weak relative compactness - lies in the reduction to o(Z,extBZ+)-relative countable compactness as opposed to o(Z,extBZ*)-relative sequential compactness. 2. There exist by now further proofs and extensions of Theorem 4.1.2 by Khurana [36] and Tweddle [731, a further simplification of Khurana’s proof by I. Namioka (personal communication), and a deduction of result 4.1.2 from Ramsey-type theorems by C. Stegall (oral communication).
68
W. Ruess
A l s o , it o u g h t t o be n o t e d t h a t f o r convex s u b s e t s H , b e e n known, see D e Wilde [ 6 1 .
4.1.2
has
According t o t h e d i s c u s s i o n s o f a r , one way t o c h a r a c t e r i z e weak c o m p a c t n e s s i n K *(X*,Y) i s t o d e t e r m i n e t h e form of t h e e x t r e m e p o i n t s i n t h e d u a l u n i t W b a l l of %* (X* , Y ) and t h e n t o a p p l y 4 . 1 . 1 and 4 . 1 . 2 . T h i s a p p r o a c h and t h e s u b s e q u e n t r e s u l t s of s e c t i o n 4 . 1 a r e t o b e f o u n d i n F l o r e t [ 1 5 , s e c t i o n s 8.10-8.131, and i n [ 4 , s e c t i o n 21. 4.1.3 Observation:
,*my*
extB (Kw*(X*, y I I * c fextBX*l@ (extBy*)
,
where
a c t s on hEKw*(X*,Y) i n t h e c a n o n i c a l way: x * @ y * ( h i = f h x * , y * ) . Kw* (X*,Y) i s a c l o s e d l i n e a r s u b s p a c e of C(BX*XBy*), BX*
Proof:
-
and BY* b e i n g endowed w i t h t h e i r r e s p e c t i v e w e a k * - t o p o l o g i e s :
-
Kw,(X*,Y)
'
C(BX*XBy*) I(x*,y*)t-t(hx*,y*)).
h
* i s o f t h e form (Kw* (X*,Y)) w i t h (x*,y*)EBX*xBy*. However, i f w e as-
Thus, e v e r y e x t r e m e p o i n t of B 6 ( x * , y * ) IKw,(X*,Y)
sume x***
=
x*@*
t o be an e x t r e m e p o i n t of B
t h e n , by t h e b i (Kw*(X* , Y ) )* l i n e a r i t y of t h e map (x*,y*)x*@y*, x* and y* n e c e s s a r i l y must b e e x t r e m e p o i n t s of B * a n d By*, r e s p e c t i v e y . T h i s p r o v e s t h e asX sertion.
I t i s now e a s y t o a r r i v e a t t h e f o l l o w i n g weak c o m p a c t n e s s r e s u l t s , c f . F l o r e t [ 1 5 , 8 . 1 0 - 8 . 1 3 1 , and [ 4 , s e c t i o n 2
4.1.4
(a)
Theorem:
A bounded sequence
hEKw,(X*,YI
i n Kw,(X*,Y)
i f and onZy i f ( h n x * , y * ) ,
(x*, y * ) E e x t B X , x e x t B Y *.
( b ) A bounded s u b s e t H o f K w * ( X * , Y )
eonverges weakly t o
c o n v e r g e s t o ( h x * , y * ) for a l l
i s weakly r e l a t i v e l y compact i f
and o n l y i f i t i s e x t B X * x e x t B * - w e a k - o p e r a t o r - t o p o l o g y
Y
countably compact.
relatively
W e n o t e t w o p a r t i c u l a r cases.
4.1.5
Corollary:
( a ) A bounded s e q u e n c e
( k n j n i n K(X,Y)
converges weakly t o k€K(X,Y)
if and o n l y i f ( k ; * x * * , y * ) n c o n v e r g e s t o ( k * * x * * , y * ) (x**, y*)EextB **xextBy*. X ( b ) A bounded s e q u e n c e weakZy t o F E C ( K , X ) F ( t ) for a l l t E K .
(Fnjn
i n CIK,XI,
for all
K compact H a u s d o r f f ,
i f and o n l y i f ( F n ( t ) ) n
converges weakly
converges ( i n Xl t o
Geometry o f operator spaces
69
For related weak compactness results, compare Kalton 135,section 21, and Tsitsas [711. After all, we can summarize that the reduction to the o(Z,extB * ) topology actually led to the desired description of weak compactngss properties by means of the corresponding pointwise-weak compactness properties. Finally, we derive a further result on the dual of Kw,(X*,Y) by combining observation 4.1.3 with a result of Haydon [21,Prop.3.1]. 4.1.6 Theorem ([52,Thm.1.7]): L e t H be a c l o s e d l i n e a r s u b s p a c e of Kw,(X*,Y), c o n t a i n i n g X s , Y . A s s u m e t h a t H d o e s n o t c o n t a i n Z 1 . T h e n H* i s a q u o t i e n t o f X * S T Y * . Proof:
The map
j : X* 5, Y*-H*
of 2.5 in section 2 is continuous
linear with I l j I ( = 1. Since H $ill Haydon’s result [21,Prop.3.1] tells us that B * = norm-clco (extBH*). We conclude, using observation4.1.3, H that BH* cnorm-clco (extBX*@extB * ) c norm-cl (j( B * Y x aT Y * ) ) cBH*, so that BH* = norm - cl (j(B * - y*)). Banach’s classical homomorphism ~~
theorem now reveals that j is a quotient map.
ExarnpZe: The James Tree space JT turned out towbe an example for a Banach space Z not containing 2, but such that Z B E Z does contain l l . Yet, according to Theorem 3.3.1 in section 3, JTZ,JT ought to be very close to not contain 21. In this sense, the James Tree space now also marks the limits of Theorem 4.1.6: The map
j : JT*S,,JT*-(JT&
4.2 THE EXTREME POINTS OF B
JT)* i s
not s u r j e c t i v e (1521).
(Kw* (X*,Y))*
The observation made in section 4.1 that extB(K *(x*,y))* is con-
tained in extB *@extBy* naturally leads to the queEtion whether there X is actually equality between the two sets, i.e. whether it is true that for x* and y* extreme points in BX+ and By*, respectively, the This problem functional x*@y* is an extreme point of B (K * (X*,Y))* ' had been solved in special cases by Singer r621 for C(K,X), Brosowski and Deutsch [ 3 1 and Strobele [671 for Co(S,X), S locally compact Haus-
dorff, and, for general completed injective tensor products X % E Y I by Hulanicki / Phelps 1311 and Tseitlin [691. C. Stegall and I [ 5 2 1 took up the general K (X*,Y)-case and proved the following result which W* shows that the extreme points in the dual unit ball of the operator space K ,(X*,Y) in fact are completely determined by the extreme points W in the duals of X and Y.
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W. Rums
4.2.1 Theorem ([52,Thm.1.1]): L e t H b e any Z i n e a r s u b s p a c e of T h e n e x t B H * = ( e x t BX * ) @ ( e x t B y * l . Kw,(X*,YI, containing X @ Y . 4.2.2 Corollary ([52,Thm.1.3]): L e t H be any l i n e a r subspace o f K(X, Yl, c o n t a i n i n g t h e f i n i t e - r a n k o p e r a t o r s . T h e n we h a v e : extBH, = (extB **lQ(extBy*l. X (For the special case of K(Z2), see Holub [30,Thm.3.1].) We established Theorem 4.2.1 in [52] by proving the following more general fact. Given X IBY c H c Kw* (X*,Y), h*EH* , Ilh*II=I , and Cx6,yZ) E extBX*XextBy*, such that h* I X B Y = 6 IX@Y , consider the set (x;5I Yb 1 M(BX*xBy*). Then we have:
(a) C is the set of norm-one extensions of h* to C(BX*xBy*), and (b) C is an extremal subset of BM(B *XB * ) . X Y Here, we give a direct proof of Theorem 4.2.1 based on Tseitlin’s result (thanks to S. Heinrich and W . Schachermayer). P r o o f of Theorem 4.2.1: Let ( x ~ , y ~ ) E e x t*xextBy*. B Then, according X to Tseitlin’s result, T = x*@y*IX%, Y E extB ( x s E y y * .Consider the set 0
0
(h* H* I h* IXgi,Y = T and Ilh*ll=I]. Then 0 is a weak*-compact convex extremal subset of BH*, and thus is the weak*-closed convex hull of its extreme points. Let htEext0. Then, since Q, is an extremal subset of BH*, h* is an extreme point of BH*, and thus, according to obser0 with x: and y: extreme vation 4.1.3 above, is of the form h* = x;@yT 0 points in B t and By*, respectively. Since h:0, we have: X = x:@y:IX@Y. This implies that x;@y: = x : % y : x;@yiIX@Y = h:IXOY is an extreme point of BH*. even on H. We conclude that 0 = ( x : @ y : ] @ =
5. GEOMETRY OF THE DUAL UNIT BALL OF Kw* (X*,Y) AND DIFFERENTIABILITY OF THE NORM After discovering that the extreme points of the dual unit ball of K *(X*,Y) are made up exactly by the set of functionals x * W * with x? and y* extreme points in the dual unit balls of X and Y, respectively, it was only natural to ask whether analogous results hold for the various special classes of extreme points. C. Stegall and I thus investigated the form of the [w*-]exposed and [w*-]strongly exposed points of the dual unit ball of Kw*(X*,Y). And, finally, Joe Diestel posed the problem of determining the denting points. We shall present in this section the respective results and discuss various applications. First, we briefly recall the definitions of these special classes of extreme points. For a detailed discussion and equivalent definitions,
Geometry of operator spaces
71
t h e reader is. r e f e r r e d t o J o e D i e s t e l ' s L e c t u r e N o t e s [ 9 1 a n d , concerning denting points, t o Phelps [ 4 4 ] . Definition:
L e t Z be a Banach s p a c e , and z o E Z w i t h I I z J l
=
1.
( a ) ( i ) z o i s a n e x p o s e d p o i n t of B Z i f t h e r e e x i s t s zzEZ*, I l z ~ l I = ,I s u c h t h a t z;(zo)=l and z* ( z ) < 1 f o r a l l zEBZ { z o ] . 0
The s e t of e x p o s e d p o i n t s of B Z w i l l b e d e n o t e d by e x p B Z .
(ii) I f Z = X* and t h e e x p o s i n g f u n c t i o n a l z*0 h a p p e n s t o b e a n e l e i s s a i d t o be a # * - e x p o s e d p o i n t of Bx* * 0 0 0 The s e t o f a l l s u c h p o i n t s w i l l b e d e n o t e d by w * - e x p B X t .
ment of X : z*=x EX, t h e n z (b)(i)
I1 z;lI=l
,
z
0
i s a s t r o n g l y e x p o s e d p o i n t of B
such t h a t
,
z;(zn) -1
2 ;
a n d , whenever
(zo)=l
t h e n llzn-zoll-O.
Z
i f t h e r e e x i s t s z:EZ*,
(zn ) n i n B Z i s s u c h t h a t
The s e t o f s t r o n g l y e x p o s e d p o i n t s of BZ w i l l b e d e n o t e d by s e x p B Z .
happens t o ( i i ) I f Z = X* and t h e s t r o n g l y e x p o s i n g f u n c t i o n a l b e an e l e m e n t of X : z * = x E X , t h e n zo i s s a i d t o b e a w * - s t r o n g l y 0
0
e x p o s e d p o i n t of BX*. The s e t of a l l s u c h p o i n t s w i l l b e d e n o t e d by w*-sexpB
*.
X ( c )( i ) z o i s a d e n t i n g p o i n t of BZ i f , g i v e n any E > 0 , t h e r e e x i s t a n d , whenever I l z l 1 1 1 , 6 > 0 and zZEZ*, I I z ~ l l = I , s u c h t h a t z;(z ) > 1 - 6 0 a n d z ; ( z ) > I - & , t h e n 1 I z - z 1 1 < E ( i . e . zo i s c o n t a i n e d i n s l i c e s o f 0
a r b i t r a r i l y s m a l l diameter).
The s e t of a l l d e n t i n g p o i n t s of B
Z ( i i ) I f Z = X* and t h e s l i c e s o f
b e g e n e r a t e d by e l e m e n t s z r
w*-denting
=
w i l l b e d e n o t e d by d e n t B Z .
( c ) ( i )c a n a l w a y s b e c h o s e n t o
x E i n X , t h e n z o i s s a i d t o be a
p o i n t o f EX*. The s e t of a l l s u c h p o i n t s w i l l b e d e n o t e d
w*-dentBX*.
by
5.1 Theorem 1 5 3 1 :
L e t H b e a c l o s e d l i n e a r s u b s p a c e of K w , ( X * , Y I ,
c o n t a i n i n g X % , Y . Then w e have:
(a)
[ w * - ] e s p BH * = [ w * - I e x p B X * @ [ w * - ] e x p BY *.
ibl
[w*-IsexpB H * = ~ w * - l s e x p B X ~ Q [ w * - l s e x p B ~ ~ .
(ci
[ w * - 1 d e n t B H * = [ w * - I d e n t B X * @ Iw * - l d e n t B y * . ( F o r t h e w*-part o f p r o p o s i t i o n
( a ) , compare J . A .
Johnson [ 3 2 1 . )
Thus, i n t h e v e i n of o u r g e n e r a l theme s e t a t t h e b e g i n n i n g of t h i s p a p e r , i n a l l cases, t h e geometry of t h e d u a l u n i t b a l l s o f X and Y c o m p l e t e l y d e t e r m i n e s t h e g e o m e t r y o f t h e , d u a l u n i t b a l l of t h e o p e r a t o r s p a c e K w J , ( X ,Y). A l t h o u g h t h e r e s u l t s of Theorem 5 . 1 q u a l i t a t i v e l y a r e a l l of t h e same n a t u r e , t h e r e s p e c t i v e methods of p r o o f t h e y r e q u i r e d t u r n e d o u t t o be q u i t e d i f f e r e n t from e a c h o t h e r and r a t h e r i n v o l v e d , s o t h a t it
72
W. Rues
would go beyond proof. Instead, general results We shall now
the scope of this survey to present even the ideas of the reader is referred to our paper L531, where more are derived. Some of these will be quoted below. discuss some consequences.
5.2 SOME EXTENSIONS AND CONSEQUENCES We first use Theorem 5.1 (b) to derive Feder / Saphar’s extension of Schatten’s result that, for H infinite-dimensional Hilbert, K ( H ) is not isometric to a dual space. 5.2.1 Theorem (Feder / Saphar [14,Thm.2]):
Assume t h a t X and Y
a r e r e f l e x i v e , and Z e t H b e any c l o s e d l i n e a r s u b s p a c e of K ( X , Y I , taining the finite-rank
operators.
I f H i s i s o m e t r i c t o a duaZ s p a c e ,
Proof:
con-
then H i s r e f l e x i v e .
Assume that H = Z*. According to our assumptions, Z** = H*
has RNP (Theorem 3.2.1 above).Hence, we have (taking into account that X and Y are reflexive): BZ** = norm-clco (sexpB * * ) . We now use z Theorem 5.1 (b) (and the fact that X and Y are reflexive) to conclude that BZ** = norm-clco (sexpBZ**) = norm-clco (sexpBH*) = = norm-cl co ( sexpBX**OsexpBy*) = = =
norm-clco (w*-sexpB **@w*-sexpBy*) = X norm-clco (w*-sexpBH*) = norm-clco (w*-sexpB * * ) =
z
norm-clco (sexpB ) = BZ . z (We used the fact that, for a general Banach space W, w*-sexpBW **= = sexpB . ) This shows that Z, and thus H, is reflexive. W =
To deduce further consequences, we first recall a classical result by Smul’yan connecting the exposed point structure of the dual unit ball with differentiability properties of the norm. Theorem ( Smul’yan [64,651 ) : L e t z o b e a n e l e m e n t of norm o n e of a Banach s p a c e 2. T h e n t h e norm o f 2 i s G a t e a u x - ( r e s p . F r s c h e t - l d i f f e r e n t i a b l e a t z o w i t h d i f f e r e n t i a l z,* i f a n d o n l y i f z o w * - e x p o s e s ( r e s p . w * - s t r o n g l y e x p o s e s ) BZ* a t z *
.
In our paper [531, instead of Theorem 5.1 (b), we prove a more general result which allows us to extend the assertion of this proposition to any linear subspace H even of Lw* (X*,Y). In particular, we can deduce the following result on the geometry of the dual unit balls of K(X,Y) and L(X,Y). 5.2.2 Corollary [531: L e t X*
Q
Y
t
H c L ( X , Y).
.
T h e n w*-sexpBH* = s e x P B X B w * - s e x p B y * I n p a r t i c u l a r , we h a v e : w * - ~ e x p B ~ ( ~ ,=~ w*) * s e x p B W ( X y,* , = w*-sexpB (X, y,* = s e x p B - * - s e x p B y * .
X
73
Geometry of operator spaces
Together with a result on G-differentiability that, again, is stronger than Theorem 5.1 (a), the extension of Theorem 5.1 (b) allows us to derive the following consequences. 5.2.3 Corollary [ 5 3 1 :
w i t h llz II=2=IlyolI.
L e t zo€X, yoEY,
( a ) Let XOYcHc3(X*,Y*l = LiX*,Y**). Then I I I I H i s F - d i f f e r e n t i a b z e a t zo@y if and o n l y i f II I1 IIY a r e F - d i f f e r e n t i a b l e a t xo and yo, r e s p e c t i v e l y . ( b ) Let -
XQY cHcCodd(BX*XBy*)
f(z*,-y*l
Then I1
I1 I I Y (
lb
= -j-(z*,y*)
(
f E C o d d ( B X * X B Y *):
IIx
$(-x*,y*)
and
=
).
is G - d i f f e r e n t i a b l e a t z Oy 0
ure G-differentiable
0
if and o n l y i f II I I x
and
a t xo and yo, r e s p e c t i v e l y .
For proposition (b), compare J.A. Johnson [321.
)
The extensions of Theorem 5.1 (a) and (b) can also be used to derive the results on F- and G-differentiability of the norms in K(X,Y) and L(X,Y) announced by S . Heinrich in [23]. We note here one particular case. L e t k o € K ( X , Y l w i t h IlkolI = 2 . T h e n 5.2.4 Corollary ( Heinrich ) : t h e following propositions are equivalent:
( a ) T h e norm of K ( X , Y I
is F - d i f f e r e n t i a b l e a t k o .
( b ) T h e norm of L ( X , Y )
is F-differentiable a t k o .
( c ) T h e norm of L ( X * * , Y * * )
i s F-differentiable
a t kz*.
(Actually, Heinrich [23,Cor.4.2] proved this up to the space
L(X,Y), i.e. the equivalence of (a) and (b).)
Some of the above results are significantly different from those of the preceding sections in that they transfer properties of X* and (X*,Y) but to the duals of much larger Y* not only to the dual of spaces of merely continuous inear operators. Further results in this direction can be found in the joint paper [ 5 3 1 with C. Stegall.
6. PROBLEMS
We close this paper with some of the problems that naturally arise from our discussion of the operator spaces K ( X , Y ) and Q,(X*,Y) in the previous sections. 6.1
When is it true that K(X,Y) does not contain (an isomorph of) co ?
6.2
Under which conditions does K(X,Y) have RNP ?
6.3
Is there an isomorphic version of Feder / Saphar’s result 5.2.1 above, i.e. if, for reflexive Banach spaces X and Y, the space K(X,Y) is isomorphic t o a dual space, is K(X,Y) then necessarily reflexive ?
74
W. Rues
6.4
Instead of %* (X*,Y) - and thus X Z E Y-and K(X,Y) - consider the completed projective tensor product X e T Y of Banach spaces X and Y: to what extent can the geometric and topological properties of X&Y and its dual L(X,Y*) be recovered from the corresponding properties of the factor spaces X and Y and their duals ?
6.5
To what extent can geometric and topological properties of particular operator spaces other than K(X,Y) - like p-absolutely summing, p-integral, or p-nuclear operators - be recovered from those of the factor spaces X and Y and their duals ? COMMENTS AND RELATED PROBLEMS
-
6.1 and 6.2: The general feeling is that like for reflexivity itself - for positive results on these weakenings of reflexivity for K(X,Y), the coincidence of L(X,Y) with K(X,Y) will play an important role. For the Radon-Nikodym property, this is being discussed in Diestel / Morrison [81. They showed: Theorem ( Diestel / Morrison [8] ) : S u p p o s e t h a t X* i s s e p a r a b z e o r r e f l e x i v e , a n d t h a t Y has R N P . T h e n , w h e n e v e r L ( X , Y I = K ( X , Y / ,
K (X,Y l h a s RNP. For extensions of this result, see Andrews [ l ] . Compare also the discussion of the RNP for operator spaces in Diestel / Uhl [10,Ch.VIII,p.258 For the problem of containing col results by A.E. Tong [681 and N.J. Kalton [351 indicate a strong connection with the coincidence of bounded and compact linear operators. Theorem ( Kalton [ 351 ) : S u p p o s e t h a t X has a n u n c o n d i t i o n a l f i n i t e - d i m e n s i o n a l e x p a n s i o n of t h e i d e n t i t y . T h e n , i f Y i s a n y i n f i n i t e - d i m e n s i o n a l Banach s p a c e , t h e f o l l o w i n g a r e e q u i v a l e n t : (a)
K(X,Yl $ e o
(el
K(X,YI
.
(bl
L(X,YI
i s complemented in L(X,Yl
.
= K(X,Y/
.
Thus, problems 6.1 and 6.2 bring our attention back to the "old" problem of characterizing spaces X and Y for which there exist non-compact bounded linear operators from X into Y, and to the problem, whether K(X,Y) ever is complemented in L(X,Y) in a "non-trivial" way : is it true that either L(X,Y) = K(X,Y), or K(X,Y) is uncomplemented in L(X,Y) ? For a thorough recent discussion of these latter problems, the reader is referred to J. Johnson [331.
6.3:
Problem 6.3 has earlier been raised by J.R. Retherford [48,p.10061. It leads us to the general problem of when K(X,Y) is (isometric or isomorphic to) a dual. In particular: is K(X) ever a dual ? J. Hennefeld [27,Cor.2.3] showed: I f X h a s a c o m p l e m e n t e d s u b s p a e e w i t h a n u n c o n d i t i o n a l b a s i s , t h e n K t X l is n o t i s o m o r p h i c t o a dual. And J. Johnson [33,Prop.l] proved the following result on the non-conjugacy of K(X,Y) :
75
Geometry of operator spaces I f Y has t h e bounded a p p r o x i m a t i o n p r o p e r t y ,
m e n t e d i n L(X,Y),
and K(X,Y)
i s n o t comple-
t h e n KIX,Y) i s n o t i s o m o r p h i c t o a c o m p l e m e n t e d sub-
s p a c e of a d u a l s p a c e .
Note that, again, an assumption on the relative position of K(X,Y) in L(X,Y) interferes ! For further information on the problem of non-conjugacy of K(X,Y) or L(X,Y), the reader may consult Hennefeld 1271 and J. Johnson [331. 6.4: This program particularly includes the problem of characterizing conditions on X and Y such that X%,Y has RNP, does not contain an isomorph of 11 or col or is weakly sequentially complete.
Note that this program is2uite ambitious, for it includes the case of the space L1 ( p , X ) = L1 (11) Bn X, which, just for the special case of weak sequential completeness and of characterizing weak compactness, turned out to be rather difficult. ( As far as I know, M. Talagrand finally showed that L1(p,X) is weakly sequentially complete whenever X is. ) A further particular problem is to find out under which conditions the space N(X,Y) of nuclear operators has RNP; compare the remarks following Corollary 3.2.2 in section 3 above.
.
In 1241, Heinrich showed that sexpBXgny = sexpB @sexpBy We deX rive this result in [53] as a further special case of our extended version of Theorem 5.1 in section 5 above.
6.5:
For this program, we only give a partial list of references related to this problem. Reflexivity of the space of p-absolutely summing operators (Ilp<-) is being discussed in Gordon/Lewis/Retherford 1171, and in Saphar 1561.
D.R. Lewis [38] discussed reflexivity, the RNP, and weak sequential completeness for various classes of a-tensor products, and specializations to corresponding classes of absolutely summing and integral operators. Heinrich [25] obtained conditions under which the spaces of p-integral and of p-absolutely summing operators (a) have RNP, and (b) do not contain (an isomorph of) c o . Non-containment of lI and of cOr and the RNP and weak sequential completeness for the spaces of p-nuclear and quasi-p-nuclear operators are being discussed in Makarov/Samarskii 1411. Andrews [I], too, has results on the RNP for the spaces of p-absolutely summing and of p-nuclear operators.
Finally, Heinrich [26] als_o obtained -results on the weak sequential Y [551 and of the spaces of completeness of the spaces X@gpY and X m
EP
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