On the complete separation of asymptotic structures in Banach spaces

Abstract

Let (ei)i denote the unit vector basis of ℓp, 1≤p<∞, or c0. We construct a reflexive Banach space with an unconditional basis that admits (ei)i as a uniformly unique spreading model while it has no subspace with a unique asymptotic model, and hence it has no asymptotic-ℓp or c0 subspace. This solves a problem of E. Odell. We also construct a space with a unique ℓ1 spreading model and no subspace with a uniformly unique ℓ1 spreading model. These results are achieved with the utilization of a new version of the method of saturation under constraints that uses sequences of functionals with increasing weights.