Banach spaces determined by their uniform structures

Following results of Bourgain and Gorelik we show that the spaces p, 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.

Overview