Extension and lifting of operators and polynomials

We study the problem of extension and lifting of operators belonging to certain operator ideals, as well as that of their associated polynomials and holomorphic functions. Our results provide a characterization of $\mathcal{L}_1$ and $\mathcal{L}_{\infty}$-spaces that includes and extends those of Lindenstrauss-Rosenthal \cite{LR} using compact operators and Gonz\'{a}lez-Guti\'{e}rrez \cite{GG} using compact polynomials. We display several examples to show the difference between extending and lifting compact (resp. weakly compact, unconditionally convergent, separable and Rosenthal) operators to operators of the same type. Finally, we show the previous results in a homological perspective, which helps the interested reader to understand the motivations and nature of the results presented.Comment: to appear in Mediterranean J. of Mathematic

On Lindenstrauss-Pe{\l}czy\'{n}ski spaces

In this work we shall be concerned with some stability aspects of the classical problem of extension of $C(K)$-valued operators. We introduce the class $\mathscr{LP}$ of Banach spaces of Lindenstrauss-Pe{\l}czy\'{n}ski type as those such that every operator from a subspace of $c_0$ into them can be extended to $c_0$. We show that all $\mathscr{LP}$-spaces are of type $\mathcal L_\infty$ but not the converse. Moreover, $\mathcal L_\infty$-spaces will be characterized as those spaces $E$ such that $E$-valued operators from $w^*(l_1,c_0)$-closed subspaces of $l_1$ extend to $l_1$. Complemented subspaces of $C(K)$ and separably injective spaces are subclasses of $\mathscr{LP}$-spaces and we show that the former does not contain the latter. It is established that $\mathcal L_\infty$-spaces not containing $l_1$ are quotients of $\mathscr{LP}$-spaces, while $\mathcal L_\infty$-spaces not containing $c_0$, quotients of an $\mathscr{LP}$-space by a separably injective space and twisted sums of $\mathscr{LP}$-spaces are $\mathscr{LP}$-spaces

Twisting non-commutative LpL_p spaces

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 $\mathcal M$. 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 $0\longrightarrow Y\longrightarrow X\longrightarrow Y \longrightarrow 0.$ For semifinite algebras, considering $L^p=L^p(\mathcal M,\tau)$ as an interpolation space between $\mathcal M$ and its predual $\mathcal M_*$ 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 can find (nontrivial) self-extensions of bimodules in all cases is left open

Derivation of vector-valued complex interpolation scales

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 [J.M.F. Castillo, V. Ferenczi and M. Gonz\'alez, \emph{Singular exact sequences generated by complex interpolation}, Trans. Amer. Math. Soc. 369 (2017) 4671--4708] are not necessary.Comment: to appear in Journal of Mathematical Analysis and Application

Banach spaces of almost universal complemented disposition

We introduce and study the notion of space of almost universal complemented disposition (a.u.c.d.) as a generalization of Kadec space. We show that every Banach space with separable dual is isometrically contained as a $1$-complemented subspace of a separable a.u.c.d. space and that all a.u.c.d. spaces with $1$-FDD are isometric and contain isometric $1$-complemented copies of every separable Banach space with $1$-FDD. We then study spaces of universal complemented disposition (u.c.d.) and provide different constructions for such spaces. We also consider spaces of u.c.d. with respect to separable spaces

K-Circular Matroids of Graphs

In 30's Hassler Whitney considered and completely solved the problem $(WP)$ of describing the classes of graphs $G$ having the same cycle matroid $M(G)$. A natural analog $(WP)'$ of Whitney's problem $(WP)$ is to describe the classes of graphs $G$ having the same matroid $M'(G)$, where $M'(G)$ is a matroid (on the edge set of $G$) distinct from $M(G)$. For example, the corresponding problem $(WP)'= (WP)_{\theta }$ for the so-called bicircular matroid $M_{\theta }(G)$ of graph $G$ was solved by Coulard, Del Greco and Wagner. We define the so-called {\em $k$-circular matroid} $M_k(G)$ on the edge set of graph $G$ for any non-negative integer $k$ so that $M(G) = M_0(G)$ and $M_{\theta }(G) = M_1(G)$. It is natural to consider the corresponding analog $(WP)_k$ of Whitney's problem $(WP)$ not only for $k=0$ and $k=1$ but also for any integer $k \ge 2$. In this paper we give a characterization of the $k$-circular matroid $M_k(G)$ by describing the main constituents (circuits, bases, and cocircuits) in terms of graph $G$ and establish some important properties of the $k$-circular matroid. The results of this paper will be used in our further research on the problem $(WP)_k$. In our next paper we use these results to study a particular problem of $(WP)_k$ on graphs uniquely defined by their $k$-circular matroids

On graphs uniquely defined by their kk-circular matroids

In 30's Hassler Whitney considered and completely solved the problem $(WP)$ of describing the classes of graphs $G$ having the same cycle matroid $M(G)$. A natural analog $(WP)'$ of Whitney's problem $(WP)$ is to describe the classes of graphs $G$ having the same matroid $M'(G)$, where $M'(G)$ is a matroid on the edge set of $G$ distinct from $M(G)$. For example, the corresponding problem $(WP)' = (WP)_{\theta }$ for the so-called bicircular matroid $M_{\theta }(G)$ of graph $G$ was solved by Coulard, Del Greco and Wagner. In our previous paper [arXive:1508.05364] we introduced and studied the so-called $k$-circular matroids $M_k(G)$ for every non-negative integer $k$ that is a natural generalization of the cycle matroid $M(G):= M_0(G)$ and of the bicircular matroid $M_{\theta }(G):= M_1(G)$ of graph $G$. In this paper (which is a continuation of our previous paper) we establish some properties of graphs guaranteeing that the graphs are uniquely defined by their $k$-circular matroids.Comment: arXiv admin note: text overlap with arXiv:1508.0536

The separable Jung constant in Banach spaces

This paper contains a study of the separable form $J_s(\cdot)$ of the classical Jung constant. We first establish, following Davis \cite{davis}, that a Banach space $X$ is $1$-separably injective if and only if $J_s(X)=1$. This characterization is then used for the understanding of new $1$-separably injective spaces. The last section establishes the inequality $\frac{1}{2}K(Y)J_s(X)\leq e(Y,X)$ connecting the separable Jung constant, Kottman's constant and the extension constant for Lipschitz maps, which is then used to obtain a simple proof of the equality $K(X,c_0)=e(X,c_0)$ of Kalton and a new characterization of $1$-separable injectivity

On the Bounded Approximation Property in Banach spaces

We prove that the kernel of a quotient operator from an $\mathcal L_1$-space onto a Banach space $X$ with the Bounded Approximation Property (BAP) has the BAP. This completes earlier results of Lusky --case $\ell_1$-- and Figiel, Johnson and Pe\l czy\'nski --case $X^*$ separable. Given a Banach space $X$, we show that if the kernel of a quotient map from some $\mathcal L_1$-space onto $X$ has the BAP then every kernel of every quotient map from any $\mathcal L_1$-space onto $X$ has the BAP. The dual result for $\mathcal L_\infty$-spaces also hold: if for some $\mathcal L_\infty$-space $E$ some quotient $E/X$ has the BAP then for every $\mathcal L_\infty$-space $E$ every quotient $E/X$ has the BAP.Comment: To appear in Israel Journal of Mathematic

On isomorphically polyhedral L∞\mathcal L_\infty-spaces

We show that there exist $\mathcal L_\infty$-subspaces of separable isomorphically polyhedral Lindenstrauss spaces that cannot be renormed to be a Lindenstrauss space.Comment: To appear in Journal of Functional Analysi