Twisted Hilbert spaces defined by bi-Lipschitz maps

Abstract

We obtain an infinite-dimensional cone of singular twisted Hilbert spaces $Z(\ddot{I}†)$ 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 $Z(\ddot{I}†)$, which gives a partial answer to a conjecture of F. Cabello Sánchez and J. Castillo.