Coarse embeddings into superstable spaces

Abstract

Krivine and Maurey proved in 1981 that every stable Banach space contains almost isometric copies of lp, for some p ∊ [1, ∞). In 1983, Raynaud showed that if a Banach space uniformly embeds into a superstable Banach space, then X must contain an isomorphic copy of lp, for some p ∊ [1, ∞). In these notes, we show that if a Banach space coarsely embeds into a superstable Banach space, then X has a spreading model isomorphic to lp, for some p ∊ [1, ∞). In particular, we obtain that there exist reflexive Banach spaces which do not coarsely embed into any superstable Banach space.