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This monograph contains a detailed exposition of the up-to-date theory of separably injective spaces: new and old results are put into perspective with concrete examples (such as l∞/c0 and C(K) spaces, where K is a finite height compact space or an F-space, ultrapowers of L∞ spaces and spaces of universal disposition).

It is no exaggeration to say that the theory of separably injective Banach spaces is strikingly different from that of injective spaces. For instance, separably injective Banach spaces are not necessarily isometric to, or complemented subspaces of, spaces of continuous functions on a compact space. Moreover, in contrast to the scarcity of examples and general results concerning injective spaces, we know of many different types of separably injective spaces and there is a rich theory around them. The monograph is completed with a preparatory chapter on injective spaces, a chapter on higher cardinal versions of separable injectivity and a lively discussion of open problems and further lines of research.

Inhaltsverzeichnis

Frontmatter

Chapter 1. A Primer on Injective Banach Spaces

To put in a proper context the results in this monograph it will be useful to keep in mind the theory of injective spaces and the general theory of \(\mathcal{L}_{\infty }\)-spaces. In this way one can compare the stability properties and the variety of examples of separably injective spaces that will be presented later with those of injective spaces. For the convenience of the reader, in this preparatory chapter we have summarized the basic properties and examples of injective Banach spaces, as well as a few remarkable examples of non-injective spaces, and some criteria that allow one to check whether a space is or is not injective. The results in this chapter have been known for many years and thus proofs will be sketched or omitted. For alternative expositions we refer to [1, Sect. 4.​3], [83, Appendix D], [194, Sects. 7–9] and [253, Sect. 2].
Antonio Avilés, Félix Cabello Sánchez, Jesús M. F. Castillo, Manuel González, Yolanda Moreno

Chapter 2. 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.
Antonio Avilés, Félix Cabello Sánchez, Jesús M. F. Castillo, Manuel González, Yolanda Moreno

Chapter 3. Spaces of Universal Disposition

In this chapter we deal with Banach spaces of universal disposition and almost universal disposition. These notions were introduced in the sixties by Gurariy, who constructed the (unique, up to isometries) separable Banach space of almost universal disposition for finite dimensional spaces in [118]. Spaces of universal disposition for separable Banach spaces are interesting for us because they are 1-separably injective (Theorem 3.5). More yet, the only way we know of obtaining separably injective p-Banach spaces is to construct p-Banach spaces of universal disposition (see Sect. 3.4.3).
Antonio Avilés, Félix Cabello Sánchez, Jesús M. F. Castillo, Manuel González, Yolanda Moreno

Chapter 4. Ultraproducts of Type $$\mathcal{L}_{\infty }$$

The Banach space ultraproduct construction is perhaps the main bridge between model theory and the theory of Banach spaces and its ramifications. Ultraproducts of Banach spaces, even at a very elementary level, proved very useful in local theory, the study of Banach lattices, and also in several nonlinear problems, such as the uniform and Lipschitz classification of Banach spaces. We refer the reader to Heinrich’s survey paper [126] and Sims’ notes [234] for two complementary accounts. Traditionally, the main investigations about Banach space ultraproducts have focused on the isometric theory, reaching a quite coherent set of results very early, as can be seen in [132]. We will review some results on the isometric theory of ultraproducts in Sect. 4.7.4, but most of the Chapter is placed in the isomorphic context.
Antonio Avilés, Félix Cabello Sánchez, Jesús M. F. Castillo, Manuel González, Yolanda Moreno

Chapter 5. $$\aleph$$ -Injectivity

Many of the results presented in this monograph about (universal) separable injectivity can be formulated in terms of the extension of operators with separable range. It is natural to attempt to obtain analogous results under more relaxed conditions in the size of the range of the operators. In this Chapter we consider the notions of (universal) \(\aleph\)-injectivity obtained by allowing domain or ranges of operators to have larger density characters. As we shall see, some results easily generalize to the higher cardinal context, some present many difficulties, and some are simply impossible. And, of course, cardinal assumptions are necessary.
Antonio Avilés, Félix Cabello Sánchez, Jesús M. F. Castillo, Manuel González, Yolanda Moreno

Chapter 6. Open Problems

In this chapter we present and discuss in some detail problems that we encountered in the course of our work. Some of them have already been mentioned in previous chapters, others have appeared under different disguises and a few are new. The contents of the sections may freely overlap.
Antonio Avilés, Félix Cabello Sánchez, Jesús M. F. Castillo, Manuel González, Yolanda Moreno

Backmatter

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