Banach spaces determined by their uniform structures

William B. Johnson; Joram Lindenstrauss; Gideon Schechtman

paper

Banach spaces determined by their uniform structures

arXiv:math/9701203

Abstract

Following results of Bourgain and Gorelik we show that the spaces $ℓ_{p}$ , $1 < p < \infty$ , as well as some related spaces have the following uniqueness property: If $X$ is a Banach space uniformly homeomorphic to one of these spaces then it is linearly isomorphic to the same space. We also prove that if a $C (K)$ space is uniformly homeomorphic to $c_{0}$ , then it is isomorphic to $c_{0}$ . We show also that there are Banach spaces which are uniformly homeomorphic to exactly $2$ isomorphically distinct spaces.