This paper deals with the following types of problems: Assume a Banach space X has some property (P). Can it be embedded into some Banach space Z with a finite dimensional decomposition having property (P), or more generally, having a property related to (P)? Secondly, given a class of Banach spaces, does there exist a Banach space in this class, or in a closely related one, which is universal for this class?
In the spirit of the classical Banach-Stone theorem, we prove that if K and S are intervals of ordinals and X is a Banach space having non-trivial cotype, then the existence of an isomorphism T from C(K, X) onto C(S,X) with distortion strictly less than 3 implies that some finite topological sum of K is homeomorphic to some finite topological sum of S. Moreover, if Xⁿ contains no subspace isomorphic to for every n ∈ ℕ, then K is homeomorphic to S. In other words, we obtain a vector-valued Banach-Stone...
We introduce a notion of disjoint envelope functions to study asymptotic structures of Banach spaces. The main result gives a new characterization of asymptotic- spaces in terms of the -behavior of “disjoint-permissible” vectors of constant coefficients. Applying this result to Tirilman spaces we obtain a negative solution to a conjecture of Casazza and Shura. Further investigation of the disjoint envelopes leads to a finite-representability result in the spirit of the Maurey-Pisier theorem.
We construct, under Axiom ♢, a family of indecomposable Banach spaces with few operators such that every operator from into is weakly compact, for all ξ ≠ η. In particular, these spaces are pairwise essentially incomparable. Assuming no additional set-theoretic axiom, we obtain this result with size instead of .
For each ordinal α < ω₁, we prove the existence of a Banach space with a basis and Szlenk index which is universal for the class of separable Banach spaces with Szlenk index not exceeding . Our proof involves developing a characterization of which Banach spaces embed into spaces with an FDD with upper Schreier space estimates.
For each ordinal α < ω₁, we prove the existence of a separable, reflexive Banach space W with a basis so that Sz(W), which is universal for the class of separable, reflexive Banach spaces X satisfying Sz(X), .
It is shown that for every k ∈ ℕ and every spreading sequence eₙₙ that generates a uniformly convex Banach space E, there exists a uniformly convex Banach space admitting eₙₙ as a k+1-iterated spreading model, but not as a k-iterated one.