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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]