This is an announcement for the paper "Rethinking Polyhedrality for Lindenstrauss Spaces" by Emanuele Casini, Enrico Miglierina, and Lukasz Piasecki.
Abstract: A recent example by the authors (see arXiv:1503.09088 [math.FA]) shows that an old result of Zippin about the existence of an isometric copy of $c$ in a separable Lindenstrauss space is incorrect. The same example proves that some characterizations of polyhedral Lindenstrauss spaces, based on the result of Zippin, are false. The main result of the present paper provides a new characterization of polyhedrality for the preduals of $\ell_{1}$ and gives a correct proof for one of the older. Indeed, we prove that for a space $X$ such that $X^{*}=\ell_{1}$ the following properties are equivalent: (1) $X$ is a polyhedral space; (2) $X$ does not contain an isometric copy of $c$; (3) $\sup\left{ x^{*}(x),:, x^{*}\in\mathrm{ext}\left(B_{X^{*}}\right)\setminus D(x)\right} <1$ for each $x\in S_{X}$, where $D(x)=\left{ x^{*}\in S_{X^{*}}:x^{*}(x)=1\right}$. By known theory, from our result follows that a generic Lindenstrauss space is polyhedral if and only if it does not contain an isometric copy of $c$. Moreover, a correct version of the result of Zippin is derived as a corollary of the main result.
Archive classification: math.FA
Mathematics Subject Classification: 46B04, 46B20, 46B25
Submitted from: enrico.miglierina@unicatt.it
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1506.08559
or