A universal reflexive space for the class of uniformly convex Banach spaces

We show that there exists a separable reflexive Banach space into which every separable uniformly convex Banach space isomorphically embeds. This solves problems raised by J. Bourgain [B] in 1980 and by W. B. Johnson in 1977 [Jo]. We also give intrinsic characterizations of separable reflexive Banach spaces which embed into a reflexive space with a block q-Hilbertian and/or a block p-Besselian finite dimensional decomposition. Dedicated to the memory of V. I. Gurarii Research supported by the National Science Foundation.