Banach spaces determined by their uniform structures

Original · EN

Following results of Bourgain and Gorelik we show that the spaces ℓₚ, 1<p<∞, 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 c₀, then it is isomorphic to c₀. We show also that there are Banach spaces which are uniformly homeomorphic to exactly 2 isomorphically distinct spaces.