This is an announcement for the paper "The Metric Approximation Property and Lipschitz-Free Spaces over Subsets of $\mathbb{R}^N$" by Eva Pernecka and Richard J. Smith.

Abstract: We prove that for certain subsets $M \subseteq \mathbb{R}^N$, $N \geqslant 1$, the Lipschitz-free space $\mathcal{F}(M)$ has the metric approximation property (MAP), with respect to any norm on $\mathbb{R}^N$. In particular, $\mathcal{F}(M)$ has the MAP whenever $M$ is a finite-dimensional compact convex set. This should be compared with a recent result of Godefroy and Ozawa, who showed that there exists a compact convex subset $M$ of a separable Banach space, for which $\mathcal{F}(M)$ fails the approximation property.

Archive classification: math.FA

Mathematics Subject Classification: 46B20, 46B28

Submitted from: richard.smith@maths.ucd.ie

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

http://front.math.ucdavis.edu/1501.07036

or

http://arXiv.org/abs/1501.07036