On uniformly finitely extensible Banach spaces (Q2260356)

This work continues an ongoing project, studying the relationships between a variety of extension properties for Banach spaces. The spaces of the title are an interesting class of objects, abbreviated in this paper as UFOs. It is shown that the properties compactly extensible, uniformly compactly extensible, and UFO, are equivalent. Moreover, a Banach space with these properties also has the uniform approximation property. Under some mild additional assumptions, it is shown that a Banach space in which every subspace is UFO must be a Hilbert space. Many of the known examples of hereditarily indecomposable spaces are shown not to be automorphic.