Complex interpolation and twisted twisted Hilbert spaces

Félix Cabello Sánchez, Jesús M. F. Castillo and Nigel J. Kalton

Vol. 276 (2015), No. 2, 287–307

DOI: 10.2140/pjm.2015.276.287

Abstract

We show that Rochberg’s generalized interpolation spaces ${\mathcal{X}}^{\left(n\right)}$ arising from analytic families of Banach spaces form exact sequences $\mathfrak{0}\to {\mathcal{X}}^{\left(n\right)}\to {\mathcal{X}}^{\left(n+k\right)}\to {\mathcal{X}}^{\left(k\right)}\to \mathfrak{0}$. We study some structural properties of those sequences; in particular, we show that nontriviality, having strictly singular quotient map, or having strictly cosingular embedding depend only on the basic case $n=k=\mathfrak{1}$. If we focus on the case of Hilbert spaces obtained from the interpolation scale of ${\ell }_p$ spaces, then ${\mathcal{X}}^{\left(\mathfrak{2}\right)}$ becomes the well-known Kalton–Peck space $Z_{\mathfrak{2}}$; we then show that ${\mathcal{X}}^{\left(n\right)}$ is (or embeds in, or is a quotient of) a twisted Hilbert space only if $n=\mathfrak{1},\mathfrak{2}$, which solves a problem posed by David Yost; and that it does not contain ${\ell }_{\mathfrak{2}}$ complemented unless $n=\mathfrak{1}$. We construct another nontrivial twisted sum of $Z_{\mathfrak{2}}$ with itself that contains ${\ell }_{\mathfrak{2}}$ complemented.

Keywords

Complex interpolation, twisted sums of Banach spaces

Mathematical Subject Classification 2010

Primary: 46B20, 46B70, 46M18, 46M35

Milestones

Received: 1 October 2013

Accepted: 22 August 2014

Published: 15 July 2015