Complexity and homogeneity in Banach spaces

Abstract

We provide an overview of a number of results concerning the complexity of isomorphism between separable Banach spaces. We also include some new results on the lattice structure of the set of spreading models of a Banach space.

Banach spaces without minimal subspaces

“…In particular, in the second case we have well-ordered chains of length ω 1 and also R-chains. This completes the picture of [13].…”

Banach spaces and Ramsey Theory: some open problems

“…Problem 6. (Block homogeneous basis problem -see [27]) Let X be a Banach space with a normalized block homogeneous Schauder basis. Is it true that X is isomorphic to c 0 or ℓ p for some 1 p < +∞?…”

“…It is likely that Problem 7 has an affirmative answer. Indeed, several partial results have been obtained (see [27] and the references therein).…”

10.4115/jla.v3i0.94

“…The Effros-Borel structure on the set S(Z) of infinite-dimensional subspaces of a separable operator space Z is defined in the same way as for Banach spaces, see e.g. [11,Chapter 12], or [6]. Reasoning as in Section 2 of [6], we show that the relations of complete isomorphism, complete biembeddability, and such, defined on S(Z), are analytic equivalence relations.…”

“…Recently, there has been much progress in describing the complexity of various relations between subspaces of a given separable Banach space. The reader is referred to [4,5,6] for the known results on the relations of isomorphism, biembeddability, and more. Isometry and local equivalence (finite representability) are handled in [16] and [7], respectively.…”