α-Minimal Banach spaces

A Banach space W with a Schauder basis is said to be -minimal for some < !1 if, for any two block subspaces Z; Y W, the Bourgain embeddability index of Z into Y is at least . We prove a dichotomy that characterises when a Banach space has an - minimal subspace, which contributes to the ongoing project, initiated by W. T. Gowers, of classifying separable Banach spaces by identifying characteristic subspaces.