Note on distortion and Bourgain $\ell _1$-index
Tom 190 / 2009
Studia Mathematica 190 (2009), 147-161 MSC: Primary 46B20; Secondary 46B03. DOI: 10.4064/sm190-2-4
Relations between different notions measuring proximity to $\ell _1$ and distortability of a Banach space are studied. The main result states that a Banach space all of whose subspaces have Bourgain $\ell _1$-index greater than $\omega ^\alpha $, $\alpha <\omega _1$, contains either an arbitrarily distortable subspace or an $\ell _1^\alpha $-asymptotic subspace.