Some more twisted Hilbert spaces

TL;DR

This paper introduces three new twisted Hilbert spaces with varied properties related to being close to Hilbert spaces, including asymptotic Hilbertianity and the HAPpy property, expanding the understanding of their structure.

Contribution

The authors construct three novel twisted Hilbert spaces, providing examples with specific properties and extending classical theorems with new twisted versions.

Findings

One space is asymptotically Hilbertian but not weak Hilbert.

Two spaces are not asymptotically Hilbertian.

One space is a HAPpy space with a slowly growing isomorphism constant.

Abstract

We provide three new examples of twisted Hilbert spaces by considering properties that are "close" to Hilbert. We denote them $Z(\mathcal J)$, $Z(\mathcal S^2)$ and $Z(\mathcal T_s^2)$. The first space is asymptotically Hilbertian but not weak Hilbert. On the opposite side, $Z(\mathcal S^2)$ and $Z(\mathcal T_s^2)$ are not asymptotically Hilbertian. Moreover, the space $Z(\mathcal T_s^2)$ is a HAPpy space and the technique to prove it gives a "twisted" version of a theorem of Johnson and Szankowski (Ann. of Math. 176:1987--2001, 2012). This is, we can construct a nontrivial twisted Hilbert space such that the isomorphism constant from its $n$-dimensional subspaces to $\ell_2^n$ grows to infinity as slowly as we wish when $n\to \infty$.