Twisted Hilbert spaces defined by bi-Lipschitz maps

TL;DR

This paper constructs a family of infinite-dimensional twisted Hilbert spaces with unique duality properties using bi-Lipschitz maps, and characterizes the Kalton-Peck space within this family.

Contribution

It introduces a cone of singular twisted Hilbert spaces isomorphic to their duals but not to their conjugate duals, and characterizes the Kalton-Peck space among these.

Findings

Constructed an infinite-dimensional cone of singular twisted Hilbert spaces.

Showed the subset of bi-Lipschitz maps from [0, ∞) to ℝ is coneable.

Provided a characterization of the Kalton-Peck space within the family.

Abstract

We obtain an infinite-dimensional cone of singular twisted Hilbert spaces $Z(\varphi)$ which are isomorphic to their duals but not to their conjugate duals. We do that by showing that the subset of all bi-Lipschitz maps from $[0, \infty)$ to $\mathbb{R}$ is coneable. We also provide a characterization of the Kalton-Peck space among all twisted Hilbert spaces of the form $Z(\varphi)$, which gives a partial answer to a conjecture of F. Cabello S\'anchez and J. Castillo.