Banach spaces determined by their uniform structures

Abstract

Following results of Bourgain and Gorelik we show that the spaces ℓ p , 1<p<∞, as well as some related spaces have the following uniqueness property: IfX 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 aC(K) space is uniformly homeomorphic toc 0, then it is isomorphic toc 0. We show also that there are Banach spaces which are uniformly homeomorphic to exactly 2 isomorphically distinct spaces.