Yolanda Moreno

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.

Banach spaces of universal disposition

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.

On the Lindenstrauss-Rosenthal theorem

We present a homological principle that governs the behaviour of couples of exact sequences of quasi-Banach spaces. Three applications are given: (i) A unifying method of proof for the results of Lindenstrauss, Rosenthal, Kalton, Peck and Kislyakov about the extension and lifting of isomorphisms inc0,ι∞,ιpandLpfor 0<p≤1; (ii) A study of the Dunford-Pettis property in duals of quotients ofL∞-spaces; and (iii) New results on the extension ofC(K)-valued operators.

On isomorphically equivalent extensions of quasi-Banach spaces

Abstract We introduce the notion of isomorphically equivalent exact sequences and quasi-linear maps. We then show how this notion is closely related with the natural equivalence of some functors Ext. In particular, we make a closer inspection of the situation for certain subspaces and quotients of L p , 0 p as well as for minimal extensions of quasi-Banach spaces. The applications include a complete answer to a problem of Fuchs in the domain of quasi-Banach spaces and a categorical proof of a result of Kalton and Peck .

Complex structures on twisted Hilbert spaces

We investigate complex structures on twisted Hilbert spaces, with special attention paid to the Kalton–Peck Z2 space and to the hyperplane problem. For any non-trivial twisted Hilbert space, we show there are always complex structures on the natural copy of the Hilbert space that cannot be extended to the whole space. Regarding the hyperplane problem we show that no complex structure on ℓ2 can be extended to a complex structure on a hyperplane of Z2 containing it.

Extensions by spaces of continuous functions

We present two complementary results on the splitting of exact sequences having the form 0 → C(K) → E → X → 0. The first one characterizes the Banach spaces X such that Ext(X, C(K)) = 0 for every compact space K. The second is a nonlinear generalization of Zippins criterion for the extension of C(K)-valued operators.

The Diagonal Functors

We obtain new categorical proofs that generalize the diagonal principles introduced in Castillo and Moreno (Israel J. Math. 140:253–270, 2004) to study the automorphic and partially automorphic character of Banach spaces. We then introduce and study the automorphy index

A Primer on Injective Banach Spaces

\mathfrak a(\cdot)

Ultraproducts of Type \mathcal{L}_{\infty }

for a Banach space, showing that

Spaces of Universal Disposition

\mathfrak a(l_\infty)= \aleph_0