Banach spaces determined by their uniform structures

Following results of Bourgain and Gorelik we show that the spaces $\ell_p$, $1<p<\infty$, 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.