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: 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.Dedicated to the memory of E. GorelikThe first author was an Erna and Jacob Michael Visiting Professor, The Weizmann Institute, 1994; and partially supported by NSF DMS 93-06376 and the U.S.-Israel Binational Science Foundation. The second and third authors were participants in the Workshop in Linear Analysis and Probability, Texas A&M University; and were partially supported by the U.S.-Israel Binational Science Foundation.