Rethinking Polyhedrality for Lindenstrauss Spaces

TL;DR

This paper revises previous characterizations of polyhedrality in Lindenstrauss spaces, providing a new equivalence for spaces with predual $\, ext{l}_1$ and correcting earlier results about isometric copies of $c$.

Contribution

It offers a new characterization of polyhedrality for preduals of $\, ext{l}_1$ and corrects prior inaccuracies regarding isometric copies of $c$ in Lindenstrauss spaces.

Findings

Polyhedrality in spaces with predual $\, ext{l}_1$ is equivalent to not containing an isometric copy of $c$.

A generic Lindenstrauss space is polyhedral iff it does not contain an isometric copy of $c$.

A corrected version of Zippin's old result is established.

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.