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We study the structure of the Rochberg Banach spaces associated to the interpolation pair (ℓ ∞ , ℓ 1) at 1/2, and of the operators defined on them. [ABSTRACT FROM AUTHOR]

Abstract In this paper we combine topological and functional analysis methods to prove that a non-locally trivial quasi-linear map defined on a C (K) must be nontrivial on a subspace isomorphic to c 0. We conclude the paper with a few examples showing that the result is optimal, and providing an application to the existence of nontrivial twisted sums of ℓ 1 and c 0. [ABSTRACT FROM AUTHOR]

Abstract Let X be a Banach space and let κ (X) denote the kernel of a quotient map ℓ 1 (Γ) → X. We show that Ext 2 (X , X ⁎) = 0 if and only if bilinear forms on κ (X) extend to ℓ 1 (Γ). From that we obtain i) If κ (X) is a L 1 -space then Ext 2 (X , X ⁎) = 0 ; ii) If X is separable, κ (X) is not an L 1 space and Ext 2 (X , X ⁎) = 0 then κ (X) has an unconditional basis. This provides new insight into a question of Palamodov in the category of Banach spaces. [ABSTRACT FROM AUTHOR]

Abstract We study complex interpolation scales obtained by vector valued amalgamation and the derivations they generate. We study their trivial and singular character and obtain examples showing that the hypotheses in the main theorems of Castillo et al. (2017) [9] are not necessary. [ABSTRACT FROM AUTHOR]

A standard way to obtain extensions (resp. liftings) of operators is by making the so-called operations of push-out (resp. pull-back). In this paper we study the preservation of some classes of operators associated with an operator ideal A under push-out extensions or pull-back liftings. We show several examples of classical operator ideals whose associated classes are preserved, we prove that the preservation of those classes under push-out extension or pull-back lifting implies that the space ideal of A satisfies the 3-space property, and we derive some results for A that are useful in the study of commutative diagrams of operators. [ABSTRACT FROM AUTHOR]

The paper makes the first steps into the study of extensions (“twisted sums”) of noncommutative L p -spaces regarded as Banach modules over the underlying von Neumann algebra M . Our approach combines Kalton's description of extensions by centralizers (these are certain maps which are, in general, neither linear nor bounded) with a general principle, due to Rochberg and Weiss, saying that whenever one finds a given Banach space Y as an intermediate space in a (complex) interpolation scale, one automatically gets a self-extension 0 ⟶ Y ⟶ X ⟶ Y ⟶ 0 . For semifinite algebras, considering L p = L p ( M , τ ) as an interpolation space between M and its predual M ⁎ one arrives at a certain self-extension of L p that is a kind of noncommutative Kalton–Peck space and carries a natural bimodule structure. Some interesting properties of these spaces are presented. For general algebras, including those of type III, the interpolation mechanism produces two (rather than one) extensions of one sided modules, one of left-modules and the other of right-modules. Whether or not one may find (nontrivial) self-extensions of bimodules in all cases is left open. [ABSTRACT FROM AUTHOR]

We show that there exist L ∞ -subspaces of separable isomorphically polyhedral Lindenstrauss spaces that cannot be renormed to be a Lindenstrauss space. [ABSTRACT FROM AUTHOR]

We show that, given a nonmetrizable compact space K having ω -derived set empty, there always exist nontrivial exact sequences 0 → c 0 → E → C ( K ) → 0 . This partially solves a problem posed in several papers: Is Ext ( C ( K ) , c 0 ) ≠ 0 for K a nonmetrizable compact set? [ABSTRACT FROM AUTHOR]

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 show that is the only Banach space with unconditional basis that satisfies the equation . This partially improves an old result by Kalton and Peck. We prove that the Kalton–Peck maps are strictly singular on a number of sequence spaces, including for , Tsirelson and Schlumprecht spaces and their duals, as well as certain super-reflexive variations of these spaces. In the last section, we give estimates of the projection constants of certain finite-dimensional twisted sums of Kalton–Peck type. [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 formulate a general theory of positions for subspaces of a Banach space: we define equivalent and isomorphic positions, study the automorphy index that measures how many non-equivalent positions Y admits in X, and obtain estimates of for X a classical Banach space such as or . Then, we study different aspects of the automorphic space problem posed by Lindenstrauss and Rosenthal; namely, does there exist a separable automorphic space different from or ? Recall that a Banach space X is said to be automorphic if every subspace Y admits only one position in X; i.e., for every subspace Y of X. We study the notion of extensible space and uniformly finitely extensible space (UFO), which are relevant since every automorphic space is extensible and every extensible space is UFO. We obtain a dichotomy theorem: Every UFO must be either an -space or a weak type 2 near-Hilbert space with the Maurey projection property. We show that a Banach space all of whose subspaces are UFO (called hereditarily UFO spaces) must be asymptotically Hilbertian; while a Banach space for which both X and are UFO must be weak Hilbert. We then refine the dichotomy theorem for Banach spaces with some additional structure. In particular, we show that an UFO with unconditional basis must be either or a superreflexive weak type 2 space; that a hereditarily UFO Köthe function space must be Hilbert; and that a rearrangement invariant space UFO must be either or a superreflexive type 2 Banach lattice. [ABSTRACT FROM AUTHOR]

The Butterfly lemma we present can be considered a reiteration theorem for differentials generated from a complex interpolation process for families of Köthe spaces. The lemma will be used to clarify the effect of different configurations in the resulting differential (because although interpolation is an orientation-free process, the obtention of differentials is not) and to round off a few aspects of Kalton's interpolation theorem. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the behavior of the limit distance function defined by a nested sequence of subsets of a real Banach space X. We first present some new criteria for the non-emptiness of the intersection of a nested sequence of sets and of their ε-neighborhoods from which we derive, among other results, Dilworth''s characterization [S.J. Dilworth, Intersections of centred sets in normed spaces, Far East J. Math. Sci. (Part II) (1988) 129–136 (special volume)] of Banach spaces not containing and Marino''s result [G. Marino, A remark on intersection of convex sets, J. Math. Anal. Appl. 284 (2003) 775-778]. Passing then to the approximation of the limit distance function, we show three types of results: (i) that the limit distance function defined by a nested sequence of non-empty bounded closed convex sets coincides with the distance function to the intersection of the weak∗-closures in the bidual; this extends and improves the results in [J.M.F. Castillo, P.L. Papini, Distance types in Banach spaces, Set-Valued Anal. 7 (1999) 101-115]; (ii) that the convexity condition is necessary; and (iii) that in spaces with separable dual, the distance function to a weak∗-compact convex set is approximable by a (non-necessarily nested) sequence of bounded closed convex sets of the space. [Copyright &y& Elsevier]

We show that Ext 2 (ℓ 2 , ℓ 2) ≠ 0 in the category of Banach spaces. This solves a sharpened version of Palamodov's problem and provides a solution to the second order version of Palais problem. We also show that Ext 2 (ℓ 1 , K) ≠ 0 in the category of quasi Banach spaces, which solves the four space problem for local convexity. [ABSTRACT FROM AUTHOR]

We discuss three problems of Koszmider on the structure of the spaces of continuous functions on the Stone compact K A generated by an almost disjoint family A of infinite subsets of ω — we present a solution to two problems and develop previous results of Marciszewski and Pol answering the third one. We will show, in particular, that assuming Martin's axiom the space C (K A) is uniquely determined up to isomorphism by the cardinality of A whenever | A | < c , while there are 2 c nonisomorphic spaces C (K A) with | A | = c. We also investigate Koszmider's problems in the context of the class of separable Rosenthal compacta and indicate the meaning of our results in the language of twisted sums of c 0 and some C (K) spaces. [ABSTRACT FROM AUTHOR]

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]

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