Some version of Gowers' dichotomy for Banach spaces.
Relations between different notions measuring proximity to ℓ₁ and distortability of a Banach space are studied. The main result states that a Banach space all of whose subspaces have Bourgain ℓ₁-index greater than , α < ω₁, contains either an arbitrarily distortable subspace or an -asymptotic subspace.
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