Félix Cabello Sánchez

Separably injective banach spaces

It is no exaggeration to say that the theory of separably injective spaces is quite different from that of injective spaces. In this chapter we will explain why. Indeed, we will enter now in the main topic of the monograph, namely, separably injective spaces and their “universal” version. After giving the main definitions and taking a look at the first natural examples one encounters, we present the basic characterizations and a number of structural properties of (universally) separable injective Banach spaces. We will show, among other things, that 1-separably injective spaces are not necessarily isometric to C-spaces, that (universally) separably injective spaces are not necessarily complemented in any C-space—the separably injective part of the assertion will be shown here while the “universal” part can be found in the next chapter—and that there exist essential differences between 1-separably injective and 2-separably injective spaces.

Twisted sums with

If X is a separable Banach space, we consider the existence of non-trivial twisted sums 0 → C(K) → Y → X → 0, where K = [0,1] or ω ω . For the case K = [0, 1] we show that there exists a twisted sum whose quotient map is strictly singular if and only if X contains no copy of 1 . If K = ω ω we prove an analogue of a theorem of Johnson and Zippin (for K = [0, 1]) by showing that all such twisted sums are trivial if X is the dual of a space with summable Szlenk index (e.g., X could be Tsirelsons space); a converse is established under the assumption that X has an unconditional finite-dimensional decomposition. We also give conditions for the existence of a twisted sum with C(ω ω ) with strictly singular quotient map.

C(K)

In this paper we present a method to obtain Banach spaces of universal and almost-universal disposition with respect to a given class M of normed spaces. The method produces, among others, the only separable Banach space of almost-universal disposition with respect to the class F of finite-dimensional spaces (Gurariĭ space G); or the only, under CH, Banach space with density character the continuum which is of universal disposition with respect to the class S of separable spaces (Kubis space K). We moreover show that K is isomorphic to an ultrapower of the Gurariĭ space and that it is not isomorphic to a complemented subspace of any C(K)-space. Other properties of spaces of universal disposition are also studied: separable injectivity, partially automorphic character and uniqueness.

spaces

-algebra [7]) but lacks the approximation property.Our techniques are quite different from those of [5] and depend on certainproperties of the extension operators introduced by Nicodemi in [10]. For stablespaces (that is, for spaces isomorphic to its square) one has the following strongerresult.Theorem 2.

Banach spaces of universal disposition

For a wide class of categories of Banach spaces, we show that the existence of an (almost) transitive element implies the existence of a separable almost transitive element. We give some applications to C 0 ( L ) spaces and abstract M -spaces. Next, we prove that Woods conjecture on almost transitivity of the norm in C 0 ( L ) can be reduced to the case in which the one-point compactification of L is metrizable. We construct a simple example of transitive M -space and we show the existence of almost transitive separable M -spaces that are isomorphic to C [0, 1]. These M -spaces have a rich M -structure and they are counterexamples for several questions about centralizers.

Polynomials on dual-isomorphic spaces

We investigate the reflexivity of the isometry group and the automorphism group of some important metric linear spaces and a1gebras. The paper consists of the following sections: 1. Preliminaries. 2. Sequence spaces. 3. Spaces of measurable functions. Hardy spaces. 5. Banach algebras of holomorphic functions. 6. Frechet algebras of holomorphic functions. 7. Spaces of continuous functions.

TRANSITIVITY OF M-SPACES AND WOOD'S CONJECTURE

It is shown that every nonlinear centralizer from

Nonlinear centralizers in homology

L_p