Isomorphic Universality and the Number of Pairwise Nonisomorphic Models in the Class of Banach Spaces - Mirna Džamonja (Abstract and Applied Analysis)
We develop the framework of natural spaces to study isomorphic embeddings of Banach spaces. We then use it to show that a sufficient failure of the generalized continuum hypothesis implies that the universality number of Banach spaces of a given density under a certain kind of positive embedding (very positive embedding) is high. An example of a very positive embedding is a positive onto embedding between C(K) and CL for 0-dimensional K and L such that the following requirement holds for all h≠0 and f≥0 in C(K): if 0≤Th≤Tf, then there are constants a≠0 and b with 0≤a·h+b≤f and a·h+b≠0.