Some topological and metric properties of the space of $\ell _1$-predual hyperplanes in $c$
Volume 168 / 2022
Abstract
We study topological and metric properties of the space $(\mathcal {H},d)$, where $\mathcal {H}$ is the set of all $\ell _1$-predual hyperplanes in the space $c$ of convergent sequences, $d$ denotes the Banach–Mazur distance and we identify hyperplanes that are almost isometric. The space $c$ and its subspace $c_0$ of sequences converging to $0$ are the simplest examples of elements in $\mathcal {H}$. First we prove that $(\mathcal {H},d)$ is homeomorphic to $(K,d_{\ell _1})$ with $K=\{x \in \ell _{1}: \left \| x\right \| \leq 1$ and $x(i)\geq x(i+1)\geq 0$ for all $i\in \mathbb {N}\}$. We provide optimal lower bounds for the distortions $\| T\|\, \| T^{-1}\|$ of isomorphic embeddings $T$ from an arbitrarily chosen element of $\mathcal {H}$ into an infinite-dimensional $L_1$-predual $X$ such that $(\operatorname {ext}B_{X^*})’\subset r B_{X^*}$ for some $r \in [0,1)$. Finally, we apply the above results to construct a homotopy contracting $\mathcal {H}$ to $c_0$ along the shortest paths in $\mathcal {H}$ and we calculate their lengths. For instance, we show that the shortest path in $\mathcal {H}$ joining $c$ and $c_0$ has length $\ln 4$.
