Uniform Equivalence between Banach Spaces by Israel Aharoni and Joram Lindenstrauss

It is a well-known result of Kadec that every two separable infinite dimensional Banach spaces are homeomorphic. Also in large classes of nonseparable Banach spaces (perhaps all) the density character of a Banach space is its only topological invariant (see the book [2] for details). The situation changes considerably if we consider uniform homeomorphisms. Several results are known which prove the nonexistence of uniform homeomorphisms between certain Banach spaces of the same density character. As a matter of fact, the following problem was raised by many mathematicians: Do there exist two nonisomorphic Banach spaces which are uniformly homeomorphic? (Le. does the uniform structure of a Banach space determine its linear structure?) For a recent survey of results related to this problem see [3]. While studying the question of existence of nonlinear liftings, we found in a surprisingly simple manner an example which answers this problem. Let T be a set of the cardinality of the continuum. Then c0(T) is lipschitz equivalent to a certain closed subspace X of l^ (i.e. there is a map T from c0(r) onto X so that T and T~ satisfy a lipschitz condition). Since there is no sequence of continuous linear functionals which separate the points in c0(T), this space is not isomorphic to a subspace of /«. Let UD F be Banach spaces and let admits a lipschitz (resp. uniformly continuous) lifting if there is a lipschitz (resp. uniformly continuous) map \p: UjV—• C/so that <pi// is the identity of U/V. If such a lifting exists then C/ is Lipschitz (respectively uniformly) homeomorphic to the direct sum V © U/V. A suitable map T which establishes the homeomorphism is Tu — (utyiç(u), <p(u)). Let {Ny}y^T be a collection of subsets of the integers N so that each Ny is infinite and Ny n Ny* is finite for y # y'. Let X be the closed linear subspace of /^ spanned by c0 and the characteristic functions Xy of Ny9 y E I\ Clearly X/c0 is isometric to c0(r) with ^x7 corresponding to the natural unit vectors ey of c0(T). The map <p admits a lipschitz lifting and thus X is lipschitz equivalent of c0(T) © c0 « c0(r). Indeed, let