SOME ISOMORPHIC PREDUALS OF `1 WHICH ARE ISOMORPHIC TO c 0 Helmut Knaust Department of Mathematical Sciences University of Texas at El Paso El Paso TX 79968-0514, USA Abstract We introduce property (F S), which asserts that a Banach space has many c 0-\sub-decompositions", and show that if X is a Banach space with property (F S) and X is isomorphic to `1, then X itself is isomorphic to c 0. 1 Introduction For quite some time it has been of interest to nd out under what circumstances a separable L1-space is isomorphic to the simplest example of an L1-space, namely c 0. Ghoussoub and Johnson [3] showed that a separable L1-space, which embeds into an order continuous Banach lattice, is isomorphic to c 0. (This result is based on earlier results by Johnson and Zippin [6], and Rosenthal [9].) Using isometric methods, Godefroy and Li [5] showed that a separable L1- space, which can be renormed into an M-ideal-in-its-bidual, is also isomorphic to c 0. Godefroy, Kalton and Saphar [4, Proposition 7.8] showed that c 0 is the only isometric predual of `1, which is a u-ideal. In [5] the authors pose the question, whether an isomorphic predual of `1, which has property (u), is isomorphic to c 0. We give a partial result in this direction: Denition 1. A separable Banach space X has property (F S), if every shrinking nite-dimensional decomposition (F n ) of X has the following property: Every increasing sequence (m n ) of positive integers has a further subsequence (k n ), so that (F kn ) is a c 0 -decomposition for its closed linear span [F kn ]. We call a sequence (F n ) of subspaces of a Banach space X a c 0 -decomposition for its closed linear span, if it satises the following: There is a constant 1991 AMS Subject Classication. Primary 46B03, 46B25. 1
C so that for all N 2 N, for all (x n ) N with x n 1 and x n 2 F n for all n = 1; 2; : : :N, Our main result is x n C: Theorem 2. A Banach space X with property (F S), whose dual X is isomorphic to `1, is isomorphic to c o. Property (F S) is an FDD-version of property (S), which was introduced by Knaust and Odell [7]. A Banach space X has property (S), if every normalized weakly null sequence in X contains a c 0 -subsequence. It was shown in [7] that property (S) is equivalent to the following: Every weak Cauchy sequence of elements (x n ) 2 Ba(X) has a subsequence (x mi ), so that for some constant C, 1X i=1 jx (x mi+1 )? x (x mi )j C for all x 2 Ba(X ): (1) Thus property (S) implies property (u), where (1) holds for far out convex combinations of the sequence (x n ) instead for a subsequence. Let us remark that Bourgain and Delbaen [1] constructed an example of a separable L1-space, which satises the Schur property, and thus properties (u), (S) and (F S). Therefore it is necessary to restrict the investigation from separable L1-spaces to isomorphic preduals of `1. Our notation is standard, and can be found e.g. in [8]. 2 Proof of the Main Result It suces to show Proposition 3. Let X be a separable Banach space with a shrinking nite dimensional decomposition. Then, if X has property (F S), X has a nitedimensional decomposition, which is a c 0 -decomposition (for the whole space). We then obtain Theorem 2 as follows: Proof. Let X be an isomorphic predual of `1 with property (F S). Since `1's standard basis is boundedly complete, it follows that X itself has a shrinking basis (x n ) (see [8, pp. 8{10]). Trivially (x n ) is a shrinking FDD; thus X possesses a c 0 -decomposition (G n ) by Proposition 3. Consequently, X is isomorphic to the c 0 -sum of the spaces (G n ). An L1-space of this form is isomorphic to c 0 (see [9, pp. 102{103]). Here is the proof of Proposition 3: 2
Proof. Let (F n ) be a shrinking FDD for X. We will show that an appropriately chosen blocking of the given FDD is a c 0 -decomposition for X. Given an increasing sequence M = (m n ) of positive integers, we let (G M n ) 1 be the shrinking FDD obtained by setting (Here we let m 0 = 0.) We dene The set A(C; N) := G M n = spanff m n?1+1; F mn?1+2; : : : ; F mn g: A = M = (m n ) 1 j (GM 2n?1) 1 is a c 0 -decomposition : ( M = (m n ) 1 j N X x 2n?1 C for all x 2n?1 2 Ba(G M 2n?1) ) is a closed (and open) set for all C; N 2 N, when the innite subsets of N are endowed with [ the \ relative topology, induced by the product topology on 2N. Since A = A(C; N), A is a Ramsey set [2, 10]. C2NN2N Consequently there is a subsequence L = (l n ) N so that either M 2 A for all subsequences M L, or M 62 A for all subsequences M L. By property (F S), the second Ramsey alternative must fail: Indeed, applying property (F S) to the FDD (G L n), we nd a subsequence (n k ) so that (G L n k ) is a c 0 -decomposition. We may assume that n k+1 > n k + 1. We let M = fl n1 ; l n2?1; l n2 ; l n3?1; : : : g; and observe that M L and (G M n ) 2 A. So the rst Ramsey alternative holds. Let L 0 = fl 2 ; l 3 ; : : : g. Both L and L 0 are in A. This means that there are constants C 1 and C 2, so that for all N 2 N? x 2n?1 C 1 for all x 2n?1 2 Ba G2n?1 L ; and x 2n?1 C 2 for all x 2n?1 2 Ba G L0 2n?1 : Since G L0 2n?1 = G L 2n for all n 2, (G L n) is a c 0 -decomposition for the whole space X. 3
3 Remarks 1. Naturally the questions arise, whether `1-preduals with property (S) have property (F S), and whether property (u) implies property (S) among `1-preduals. 2. It is easy to check that c 0 has property (F S). 3. Not every separable Banach space with property (S) has property (F S); indeed we have the following example: Example 4. There is a Banach space with a shrinking unconditional basis, which satises property (S), but fails property (F S). Proof. We use a tree-space example constructed by Talagrand [11]. Let D S = 1 n=0 f0; 1gn denote a binary tree. An element ' 2 D is called a node. If ' = ( i ) n with i=1 i 2 f0; 1g we say ' has length j'j = n. As usual, we dene a partial order on D as follows: ' if, writing ' = ( i ) n and = i=1 ( i) m, i=1 n m and i = i for all i = 1; : : : ; n. If ' and are incomparable, we write ' k. For x 2 c 00, let n2n x = sup 80 >< @ X >: j'j=n sup ' 1 9 1=2>= jx( )j A 2 >; : The Banach space T is the completion of c 00 under this norm. Let (e ' ) denote the standard basis in its lexicographical order, dened by e ' ( ) = 1, if ' = and e ' ( ) = 0 otherwise. Obviously (e ' ) is a 1- unconditional basis. Since `1 does not embed into T [11], it follows that the basis is shrinking. Talagrand showed that T has property (S). We claim that T fails property (F S). Consider the nite-dimensional decomposition (F n ), dened by F n = [e ' ] j'j=n for all n 2 N: Clearly (F n ) is a shrinking decomposition for T. We show that (F n ) has no c 0 -sub-decomposition. Let (m n ) be a subsequence of N. We choose inductively a sequence of nodes (' n ). Let ' 1 be such that j' 1 j = m 1. Once ' n 's have been chosen for n = 2 k?1 ; : : : ; 2 k? 1, we proceed as follows: Let n and n denote the two distinct successor nodes of ' n, (i.e. n 6= n, n ' n, n ' n and j n j = j n j = m n + 1). Then choose '`, ` = 2 k ; : : : ; 2 k+1? 1 so that j'`j = m` for all ` = 2 k ; : : : ; 2 k+1?1 and so that ' 2n n, ' 2n+1 n for n = 2 k?1 ; : : : ; 2 k? 1. The construction ensures that ' 2n k ' 2n+1 for n = 2 k?1 ; : : : ; 2 k? 1, and that ' n = min(' 2n ; ' 2n+1 ) for n = 2 k?1 ; : : : ; 2 k? 1, i.e. whenever a node 2 D satises ' 2n and ' 2n+1, then ' n. 4
Clearly e 'n 2 F mn and ke 'n k = 1 for all n 2 N; on the other hand it is easy to see that k 2 k+1?1 X n=2 k e 'n k 2 k=2 for all k 2 N: Thus (F mn ) fails to be a c 0 -decomposition. References [1] J.Bourgain & F. Delbaen, A class of special L 1 -spaces, Acta Math. 145(1980), 155-176. [2] E.E. Ellentuck, A new proof that analytic sets are Ramsey, J. Symbolic Logic 39(1974), 163{165. [3] N. Ghoussoub & W.B. Johnson, Factoring operators through Banach lattices not containing C(0; 1), Math. Z. 194(1987), 153{171. [4] G. Godefroy, N.J. Kalton & D. Saphar, Unconditional ideals in Banach spaces, Studia Math. 104(1993), 13{59. [5] G. Godefroy & D. Li, Some natural families of M -ideals, Math. Scand. 66(1990), 249-263. [6] W.B. Johnson & M. Zippin, On subspaces of quotients of (G n )`p and (G n ) c0, Israel J. Math. 13(1972), 311-316. [7] H. Knaust & E. Odell, On c 0 sequences in Banach spaces, Israel J. Math. 67(1989), 153-169. [8] J. Lindenstrauss & L. Tzafriri, Classical Banach spaces I. Springer- Verlag, Berlin, Heidelberg, New York, 1977. [9] H.P. Rosenthal, A characterization of c 0 and some remarks concerning the Grothendieck property, pp. 95{108, in: Longhorn Notes 1982{83, The University of Texas, Austin. [10] J. Silver, Every analytic set is Ramsey, J. Symbolic Logic 35(1970), 60{64. [11] M. Talagrand, La propriete de Dunford-Pettis dans C(K; E) et L 1 (E), Israel J. Math. 44(1983), 317{321. January 12, 95 5