Some more twisted Hilbert spaces

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 $n\to \mathrm{\infty }$.