On uniform homeomorphisms of the unit spheres of certain Banach lattices

Abstract

We prove that if X is an infinite dimensional Banach lattice with a weak unit then there exists a probability space (Ω,Σ,/i) so that the unit sphere of (Li(Ω,Σ,/i)) is uniformly homeomorphic to the unit sphere S(X) if and only if X does not contain Z^'s uniformly.

The distortion problem

“…Indeed, Y. Raynaud [31] proved that if X is not reflexive and Ba(X) embeds uniformly into ℓ 2 , then X admits an ℓ 1 -spreading model. Fouad Chaatit [2] has extended Theorem 2.1. He showed one can replace the hypothesis that X has an unconditional basis with the more general assumption that X is a separable infinite dimensional Banach lattice.…”

“…Fouad Chaatit [2] has extended Theorem 2.1. He showed one can replace the hypothesis that X has an unconditional basis with the more general assumption that X is a separable infinite dimensional Banach lattice.…”

“…The 2-convexification of X in this norm, X (2) , satisfies M 2q (X (2) ) = 1 = M 2 (X (2) ) ([19, p.54]). In particular X (2) is uniformly convex and uniformly smooth ([19, p.80]) and so F X (2) : S(ℓ 1 ) → S(X (2) ) is a uniform homeomorphism by Proposition 2.6. Thus G 2 •F X (2) : S(ℓ 1 ) → S(X) is a uniform homeomorphism by Proposition 2.8.…”

Comparison of Metric Spectral Gaps

“…9, Sec. 1]); Banach spaces of finite cotype with an unconditional basis, as shown by Odell and Schlumprecht [OS94]; more generally, Banach lattices of finite cotype, as shown by Chaatit [Cha95] (see [BL00, Ch. 9, Sec.…”

On the Stability of the Differential Process Generated by Complex Interpolation

“…Analytic families of Banach spaces are relevant to other topics in Banach space theory such as the construction of uniformly convex hereditarily indecomposable spaces [19], the study of θ-Hilbertian spaces [30] introduced by Pisier, or problems about the uniform structure of Banach spaces. Recall that the question of whether the unit sphere of a uniformly convex space is uniformly homeomorphic to the unit sphere of a Hilbert space can be positively answered for Köthe spaces using interpolation methods: if X 0 and X 1 are uniformly convex spaces, then the unit spheres of X θ and X ν are uniformly homeomorphic for 0 < θ, ν < 1 by a result of Daher [16]; this fact, together with an extrapolation theorem of Pisier [30], implies that the unit sphere of a uniformly convex Köthe space is uniformly homeomorphic to the unit sphere of the Hilbert space (see also [11]). Thus, an extrapolation theorem for arbitrary uniformly convex spaces would provide a positive answer to the problem.…”