Banach spaces whose duals are isomorphic to l 1. Edward Odell

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1 1 Banach spaces whose duals are isomorphic to l 1 Edward Odell The University of Texas at Austin Richmond, Virginia November 7, 2009

2 2 The dual of c 0 is isometric to l 1 : c 0 = l 1. Many spaces have this property: If K is countable compact metric then C(K) = l 1. c 0 and C(K) are hereditarily c 0 : For all X C(K), c 0 X. Note X, Y, Z,... shall denote separable infinite dimensional Banach spaces. X Y denotes X is a closed subspace of Y. Now: Suppose X = l 1. Certain things can be said about X. [JZ] X is a quotient of C[0, 1]. [F] X is hereditarily c 0. [A] There exists such an X which is not a quotient of C(K) for any countable compact K.

3 3 A long standing open problem (30 years ago) was: If X l 1 does c 0 X? This was solved in the negative by Bourgain and Delbaen. [BD] There exists a space X which is somewhat reflexive (for all Y X, there exist a reflexive W Y) and X l 1. [H] In fact every such Y contains some l p, 1 < p <. Question Given X can we embed it into Y with Y l 1? Well, clearly not if Y is not separable. But what if Y is separable?

4 4 [LR] If X l 1 then X is L. Definition X is L,λ if there exists an increasing sequence (F n ) of finite dimensional subspaces of X with X = n F n and F n n. λ l dim F n for all Recall that C[0, 1] is L,1+ε and every X embeds into C[0, 1]. [LS] If X is L and X is separable, then X l 1. So we are asking if the family of L spaces with separable dual is universal for {Y : Y is separable}. Szlenk proved that no one space would work. Bourgain proved that X is universal for {Y : Y is reflexive} then C[0, 1] Y.

5 5 Main Theorem [FOS] Let X be separable. Then X embeds into a space Y with Y l 1. Moreover a) If c 0 X, we can also obtain c 0 Y. b) If X is reflexive, we can also obtain that Y is somewhat reflexive. The proof uses the [BD] construction and results of [FOSZ].

6 6 The [BD] construction of spaces Y with Y l 1. Y is constructed as a subspace of l with shrinking basis (d n ) (i.e., the biorthogonal functionals (d n ) are a basis for Y l 1 ). n th coordinate }{{} d n = (0, 0,..., 0, 1, x, x, x,...) l The biorthogonal functionals will be d n = (x, x,..., x, 1, 0, 0,...) l 1 Thus d n = e n c n where (e n ) is the unit vector basis of l 1 and c n span(e i )n 1 1. The trick is to define the c n s, hence d n s and let (d n ) l be biorthogonal to the d n s.

7 7 Fix 0 < θ < 1/2. The c n s are defined recursively. Clearly c 1 = 0. Assume all is defined for 1, 2,..., n 1. span(d i )n 1 1 = span(e i )n 1 1 = l 1 (n 1). So we have basis projections P 1,..., P n 1 for (d i )n 1 1 defined on l 1 (n 1). We will choose c n to be a BD-functional: c n = β(i P k )b for some β θ, b Ball l1 (k,n), 0 k < n 1. or c n = e j +β(i P k)b for β θ, 1 j k < n 1, b Ball l1 (k,n).

8 8 One can deduce from this that (d n) is a basis for l 1 and Y = [(d n )] is L. If (d n ) is shrinking (equivalently, (dn ) is boundedly complete), then Y l 1. This construction has a great deal of freedom in it! The trick is to choose the c n s to achieve what you wish.

9 9 To prove our main theorem we will construct a countable set Γ = n n, n s finite and disjoint. We will work in l (Γ) and l 1 (Γ). The notation will be such that an element γ Γ codes c γ. To start, by [FOSZ] we may assume that our space X with separable dual has an FDD(E n ) satisfying subsequential C T α,c upper estimates: Hence, for a block sequence (x i )n 1 w.r.t. (E i ) C n 1 x i n x i tk T i α,c where k i = min supp E (x i ) and (t i ) is the unit vector basis of T α,c. 1

10 10 T α,c : 1 α < ω 1, 0 < c < 1 S α = α th class of Schreier sets in [N] <ω (a certain compact, hereditary, spreading family of increasing complexity as α ) (E i ) n 1 is S α-admissible if E 1 < < E n are finite subsets of N and (min E i ) n 1 S α. For x c 00 x = x sup c (E i ) n 1 Sα-admissible (The famous Tsirelson space is T 1,1/2 ) n x Ei i=1

11 11 Γ {(x 1,..., x j ) : (x i ) j i=1 w.r.t. (E i ) and x = is a block sequence j x i B X } 1 Moreover the collection of all such x s in Γ will norm X. The idea will be to do the BD-construction so that X embeds into Y l (Γ) via a mapping Φ in a natural way: if γ = (x 1,..., x j ) and x = j 1 x i, then e γ(φ(x)) x (x). Notation: for such γ, x i=l Ei i=k range γ = range E (x ) = [k,l] if and no smaller interval satisfies this.

12 12 We order the closed intervals in N as [1] 1 [2] 2 [1, 2] 3 [3] 4 [2, 3] 5 [1, 3] 6 [4] 7 [3, 4] 8 [2, 4] 9 [1, 4] We put γ = (x 1,...,x j ) n if range(γ) = n th interval. Fix θ 1/32. The elements of Γ will be chosen so that for γ = (x 1,..., x j ), x i < θ unless x i n S E n and x i + x i+1 > θ if i < j 1. This is called a θ-decomposition of x (= j 1 x i ) and by [FOSZ] every x has such a decomposition with (min supp E x i )j 1 G, a certain compact subset of [N] <ω.

13 13 G is what ultimately yields that (d γ ) is shrinking. We need to define the Cγ s. Let γ = (x 1,..., x j ) If j = 1, Cγ = 0 If j = 2, Cγ = x 1 e η(γ) + x 2 e ξ(γ) where η(γ) = θ-decomposition of x 1 / x 1 and ξ(γ) = θ-decomposition of x 2 / x 2 If j > 2 Cγ = e (x 1,...,x j 1 ) + x j e ξ(γ) where ξ(γ) = θ-decomposition of x j / x j.

14 14 Now we need each n to be finite so we actually work with certain approximations, but we ignore that here. From the construction for γ = (x 1,..., x n), n 2, n e γ = x 1 e ξ(1) + ( x j e ξ(j) + d γ(j) ) where γ(j) = (x 1,...,x j ), ξ(j) = θ-decomposition of x j / x j. j=2 This turns out to be a θ-decomposition of e γ in Y which is G-admissible for some compact family G and yields Y l 1.

15 15 8 How does the embedding of X into Y work? Well, we had in the background certain finite ε i -nets A i S E i and if E i mi, we embed x E i into F mi span(d γ : γ mi ) by e γ (x) = e (x ) (x) = x (x) for γ mi, γ = (x ). This specifies the coordinate values of Φ(x) for γ m i 1 j x x x x x m i (Γ) The future values are determined by the future coordinates of d γ. We extend Φ linearly to span(e i ) and obtain e (x 1,...,x j )(Φ(x)) j 1 x i (x)

16 16 Φ(X) is contained in i=1f mi 1 F i = Y, where F i = span{d γ : γ i }. The F i s yield that Y is L : span{d γ : γ n i } 2 l n 1 i i=1 The not c 0 and reflexive X cases are handled with the above as the base of the construction. We augment each n to n = n θ n and use the extra coordinates in θ n to achieve the appropriate goal.

17 17 Remark It was not known if a space X with X separable and c 0 X embedded into a space Z with a shrinking basis with c 0 Z. Somehow the L structure, which gives much more, allows this to be done.

18 18 Argyros and Haydon [AH] recently used the BD-construction to produce a space Y satisfying Y is L, Y l 1 Y is H.I. [if Y = W Z then inf(dim W, dim Z) < ] L(Y) = {λi + K : K compact operator} L(Y) is amenable Every T L(Y) has an invariant subspace L(Y) is separable c 0 L(Y)

19 19 The proof uses a certain mixed Tsirelson space T(m j, n j ) j=1 n γ = { 1 x = x sup j m j n j } x Ej : E 1 < < E nj i=1 (n, 1mj, b ), b Ball l1 (n 1) C γ = 1 m j b or γ = (n,ξ, 1mj, b ), ξ p, p < n 1, b Ball l1 (p,n) C γ = e ξ + 1 m j P (p,n) b Subject to severe restrictions along the lines of HI-constructions (as in [GM])