Local complementation and the extension of bilinear mappings

TL;DR

This paper explores the extension of bilinear forms in Banach spaces, revealing limitations related to Hilbert space structures and introducing local $eta$-complementation concepts.

Contribution

It establishes new conditions under which bilinear form extensions fail and connects local $eta$-complementation with operator extendability in Banach spaces.

Findings

Non-Hilbert spaces can lack Aron-Berner extensions for some subspaces.

Extension of all bilinear forms implies the space contains no uniform $oldsymbol{ ext{l}}_p^n$ copies for p in [1,2).

Bilinear versions of classical theorems like Lindenstrauss-Pe{}czy
44ski and Johnson-Zippin fail.

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.