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Abstract: We continue the study of Uniformly Finitely Extensible Banach spaces (in short, UFO) initiated in Moreno and Plichko (2009) [39] and Castillo and Plichko (2010) [18]. We show that they have the Uniform Approximation Property of Pełczyński and Rosenthal and are compactly extensible. We will also consider their connection with the automorphic space problem of Lindenstrauss and Rosenthal – do there exist automorphic spaces other than and ? – showing that a space all whose subspaces are UFO must be automorphic when it is Hereditarily Indecomposable (HI), and a Hilbert space when it is either locally minimal or isomorphic to its square. We will finally show that most HI – among them, the super-reflexive HI space constructed by Ferenczi – and asymptotically spaces in the literature cannot be automorphic. [Copyright &y& Elsevier]

Abstract: Let denote any of the following classes of -spaces: -spaces, Lindenstrauss spaces, -separably injective spaces, universally -separably injective spaces, -Lindenstrauss–Pełczyński spaces or -spaces. We show that every Banach space can be isometrically embedded into a space so that every operator with can be extended to an operator with the same norm. [Copyright &y& Elsevier]

Abstract: We study the stability properties of the class of weak*-extensible spaces introduced by Wang, Zhao, and Qiang showing, among other things, that weak*-extensibility is equivalent to having a weak*-sequentially continuous dual ball (in short, w*SC) for duals of separable spaces or twisted sums of w*SC spaces. This shows that weak*-extensibility is not a -space property, solving a question posed by Wang, Zhao, and Qiang. We also introduce a restricted form of weak*-extensibility, called separable weak*-extensibility, and show that separably weak*-extensible Banach spaces have the Gelfand–Phillips property, although they are not necessarily w*SC spaces. [Copyright &y& Elsevier]

Abstract: We present a characterization of -spaces in terms of the continuity of -valued linear maps on dense subspaces. This provides a complete solution to a question investigated by Bogachev, Kircheim, Schachermayer and Shkarin. [Copyright &y& Elsevier]

Abstract: We deal with two weak forms of injectivity which turn out to have a rich structure behind: separable injectivity and universal separable injectivity. We show several structural and stability properties of these classes of Banach spaces. We provide natural examples of (universally) separably injective spaces, including ultraproducts built over countably incomplete ultrafilters, in spite of the fact that these ultraproducts are never injective. We obtain two fundamental characterizations of universally separably injective spaces. (a) A Banach space is universally separably injective if and only if every separable subspace is contained in a copy of inside . (b) A Banach space is universally separably injective if and only if for every separable space one has . Section 6 focuses on special properties of 1-separably injective spaces. Lindenstrauss proved in the middle sixties that, under CH, 1-separably injective spaces are 1-universally separably injective and left open the question in ZFC. We construct a consistent example of a Banach space of type which is 1-separably injective but not universally 1-separably injective. [Copyright &y& Elsevier]