Weakly Null Sequences in L1

In [MR], the second author and H. P. Rosenthal constructed the first examples of weakly null normalized sequences which do not have any unconditionally basic subsequences. Much more recently, the second author and W. T. Gowers [GM] constructed infinite dimensional Banach spaces which do not contain any unconditionally basic sequences. These later examples are of a different character than the examples in [MR]. For example, in [MR], it is shown that if K is a sufficiently complex countable compact metric space, then the space C(K) contains a weakly null normalized sequence which does not have an unconditionally basic subsequence, and it is known [PS] that every infinite dimensional subspace of such a C(K) space contains an unconditionally basic sequence, in fact, a sequence which is equivalent to the unit vector basis of c0. In [MR] it was asked if a similar example exists in L1. The space L1 is similar to C(K) in that every infinite dimensional subspace contains an unconditionally basic sequence. Indeed, if the subspace is not reflexive, then 1 embeds into the subspace [KP]. If a subspace X of L1 is reflexive, then it embeds into Lp for some 1 < p ≤ 2 [R]. Since Lp has an unconditional basis, every weakly null normalized sequence in X has an unconditionally basic subsequence. The main result in this paper, Theorem 1, is that there is a weakly null normalized sequence {fi}i=1 in L1 which has no unconditionally basic subsequence. In fact, the sequence {fi}i=1 has the stronger property that for every ε > 0, the (conditional) Haar basis is (1 + ε)-equivalent to a block basis of every subsequence of {fi}i=1. This is analogous to the result in [MR] that if K is a sufficiently complex countable compact metric space, then the space C(K) contains a weakly null normalized sequence {xn}n=1 so that every initial segment of the (conditional) summing basis is (1 + ε)-equivalent to a block basis of every subsequence of {xn}n=1. Theorem 1 can also be compared to the result of [MS] that for 1 < p < 2, there is a 1-symmetric basic sequence {gn}n=1 in Lp so that the Haar basis is equivalent to a block basis of every subsequence of {gn}n=1. In this result, there is a lower