Local complementation and the extension of bilinear mappings

Abstract

We study different aspects of the connections between local theory of Banach spaces and the problem of the extension of bilinear forms from subspaces of Banach spaces. Among other results, we prove that if $X$ is not a Hilbert space then one may find a subspace of $X$ for which there is no Aron-Berner extension. We also obtain that the extension of bilinear forms from all the subspaces of a given $X$ forces such $X$ to contain no uniform copies of $\ell_p^n$ for $p\in[1,2)$. In particular, $X$ must have type $2-\epsilon$ for every $\epsilon>0$. Also, we show that the bilinear version of the Lindenstrauss-Pe{\l}czy\'nski and Johnson-Zippin theorems fail. We will then consider the notion of locally $\alpha$-complemented subspace for a reasonable tensor norm $\alpha$, and study the connections between $\alpha$-local complementation and the extendability of $\alpha^*$ -integral operators.Comment: to appear in Mathematical Proceedings of the Cambridge Philosophical Societ

Extension of polynomials defined on subspaces

“…This result was first established in [13, théorème 3•7]. Recently, another proof has appeared in [8]. We prove it here in a different way, with a proof that follows the same pattern as the proofs of Theorems 1•3 and 1•7.…”

Polynomial ideals from a nonlinear viewpoint

“…For the last three decades, a huge amount of work has been done on polynomial ideals. The theory has been developed pari passu with the theory of ideals of multilinear operators (multi-ideals), and connections have been established with other topics, such as (just a few references are given): infinite dimensional holomorphy [8,21,43], topological tensor products [23,30,42], ultrapower stability [40,41], quantum information theory [48,53], Dirichlet series [32,34], integral formulas/stable measures [22], coherence/compatibility [20,50], Bishop-Phelps-Bollobás-Lindenstrauss circle of ideas [2,24], unconditional bases [23,33], interpolation theory [4,18,35], classical inequalities (Hardy-Littlewood, Bohnenblust-Hille, Blei) [3,4,5,35,59], summability properties [1,47,51], approximation properties [8,29,30], extension of multilinear operators and polynomials [27,42,45], Aron-Berner stability [12,42], hypercyclic convolution operators [9,19], homological methods [27], lineability/spaceability [13,…”

Representation of Extendible Bilinear Forms

“…Since then, several authors have found various Hahn-Banach extension theorems, such as in [5,12] and more recently in [4]. (See also the references therein.…”