An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree T of a certain type on a space X is presumed to have a branch with some property. It is shown that then X can be embedded into a space with an FDD (Ei) so that all normalized sequences in X which are alm...

We present a characterization of the sequences in a Banach space that have a sequence of blocks cofinally equivalent to the standard basis of `p (1 ≤ p <∞) or c0.

Let $(X,d)$ be an infinite compact metric space, let $(B,parallel . parallel)$ be a unital Banach space, and take $alpha in (0,1).$ In this work, at first we define the big and little $alpha$-Lipschitz vector-valued (B-valued) operator algebras, and consider the little $alpha$-lipschitz $B$-valued operator algebra, $lip_{alpha}(X,B)$. Then we characterize its second dual space.

A Banach space B is isometric with a subspace of its second conjugate space B** under the "natural mapping" for which the element of B** which corresponds to the element xo of B is the linear functional Fxz defined by Fxs(f) = f(xo) for each f of B*. If every F of B** is of this form, then B is said to be reflexive and B is isometric with B** under this natural mapping. The purpose of this note...

M. Gromov [7] suggested to use coarse embeddings into a Hilbert space or into a uniformly convex space as a tool for solving some of the well-known problems. G. Yu [21] and G. Kasparov and G. Yu [11] have shown that this is indeed a very powerful tool. On the other hand, there exist separable metric spaces ([6] and [5, Section 6]) which are not coarsely embeddable into Hilbert spaces. In [9] (s...

‎ Let $(X,d)$ be a metric space and $Jsubseteq (0,infty)$ be a nonempty set. We study the structure of the arbitrary intersection of vector-valued Lipschitz algebras, and define a special Banach subalgebra of $cap{Lip_gamma (X,E):gammain J}$, where $E$ is a Banach algebra, denoted by $ILip_J (X,E)$. Mainly, we investigate $C-$character amenability of $ILip_J (X,E)$.

Many questions in Banach space theory are of the type: Let X be an infinite dimensional Banach space. Let (P) be a property. Does X contain a closed infinite dimensional subspace Y with (P)? Sometimes the question takes the form: If X has a certain property (Q), does X contain Y having (P)? Of course the answers and solutions (if known) depend upon the particular properties involved. But there ...

For a Banach algebra $fA$, we introduce ~$c.c(fA)$, the set of all $phiin fA^*$ such that $theta_phi:fAto fA^*$ is a completely continuous operator, where $theta_phi$ is defined by $theta_phi(a)=acdotphi$~~ for all $ain fA$. We call $fA$, a completely continuous Banach algebra if $c.c(fA)=fA^*$. We give some examples of completely continuous Banach algebras and a suffici...

We prove that a Schauder frame for any separable Banach space is shrinking if and only if it has an associated space with a shrinking basis, and that a Schauder frame for any separable Banach space is shrinking and boundedly complete if and only if it has a reflexive associated space. To obtain these results, we prove that the upper and lower estimate theorems for finite dimensional decompositi...

We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael’s Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse o...