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 l(p)(n) for p is an element of [1, 2). In particular, X must have type 2 - epsilon for every epsilon > 0. Also, we show that the bilinear version of the Lindenstrauss-Pelczynski 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.