Complemented copies of ℓ¹ and Pelczynski's property (V*) in Bochner function spaces

1994/10/31 by Randrianantoanina, Narcisse
#46E #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.math/9410204

Abstract

Let X be a Banach space and (f_n)_n be a bounded sequence in L¹(X). We prove a complemented version of the celebrated Talagrand's dichotomy i.e we show that if (e_n)_n denotes the unit vector basis of c₀, there exists a sequence g_n ∈ conv(f_n,f_n+1,…) such that for almost every ω, either the sequence (g_n(ω) ⊗ e_n) is weakly Cauchy in X \widehat⊗_πc₀ or it is equivalent to the unit vector basis of ℓ¹. We then get a criterion for a bounded sequence to contain a subsequence equivalent to a complemented copy of ℓ¹ in L¹(X). As an application, we show that for a Banach space X, the space L¹(X) has Pełczyński's property (V^*) if and only if X does.

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