This is an announcement for the paper "Local complementation and the extension of bilinear mappings" by J.M.F. Castillo, A. Defant, R. Garcia, D. Perez-Garcia, and J. Suarez.

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 $2-\varepsilon$ for every $\varepsilon>0$. Also, we show that the bilinear version of the Lindenstrauss-Pe\l czy'{n}ski 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.

Archive classification: math.FA

Remarks: to appear in Mathematical Proceedings of the Cambridge Philosophical

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1106.5089

or

http://arXiv.org/abs/1106.5089