Equilateral and separated sets in some Hilbert generated Banach spaces

TL;DR

This paper investigates the geometric structure of certain nonseparable Banach spaces generated by Hilbert spaces, focusing on the existence of large equilateral and separated sets, and explores how set-theoretic assumptions influence these properties.

Contribution

It provides new results on the existence of uncountable equilateral and separated sets in Hilbert generated Banach spaces, highlighting the impact of the continuum hypothesis.

Findings

Under CH, some nonseparable Hilbert generated spaces lack uncountable equilateral sets.

Certain spaces contain isomorphic copies of  in every nonseparable subspace.

The existence of uncountable separated sets varies with set-theoretic assumptions.

Abstract

We study Hilbert generated versions of nonseparable Banach spaces $\mathcal X$ considered by Shelah, Stepr\=ans and Wark where the behavior of the norm on nonseparable subsets is so irregular that it does not allow any linear bounded operator on $\mathcal X$ other than a diagonal operator (or a scalar multiple of the identity) plus a separable range operator. We address the questions if these spaces admit uncountable equilateral sets and if their unit spheres admit uncountable $(1+)$-separated or $(1+\varepsilon)$-separated sets. We resolve some of the above questions for two types of these spaces by showing both absolute and undecidability results. The corollaries are that the continuum hypothesis (in fact: the existence of a nonmeager set of reals of the first uncountable cardinality) implies the existence of an equivalent renorming of the nonsepareble Hilbert space $\ell_2(\omega_1)$ which does not admit any uncountable equilateral set and it implies the existence of a nonseparable Hilbert generated Banach space containing an isomorphic copy of $\ell_2$ in each nonseparable subspace, whose unit sphere does not admit an uncountable equilateral set and does not admit an uncountable $(1+\varepsilon)$-separated set for any $\varepsilon>0$. This could be compared with a recent result by H\'ajek, Kania and Russo saying that all nonseparable reflexive Banach spaces admit uncountable $(1+\varepsilon)$-separated sets in their unit spheres.