Complex structures on twisted Hilbert spaces

TL;DR

This paper studies complex structures on twisted Hilbert spaces, especially the Kalton-Peck space, revealing conditions under which complex structures can or cannot be extended, and addressing the hyperplane problem.

Contribution

It provides new insights into the extension of complex structures on twisted Hilbert spaces and explores the hyperplane problem within this context.

Findings

Certain complex structures cannot be extended to twisted Hilbert spaces.

Rearrangement invariant K"othe function spaces allow extension of complex structures.

No complex structure on  extends to a hyperplane of the Kalton-Peck space.

Abstract

We investigate complex structures on twisted Hilbert spaces, with special attention paid to the Kalton-Peck $Z_2$ space and to the hyperplane problem. We consider (nontrivial) twisted Hilbert spaces generated by centralizers obtained from an interpolation scale of K\"othe function spaces. We show there are always complex structures on the Hilbert space that cannot be extended to the twisted Hilbert space. If, however, the scale is formed by rearrangement invariant K\"othe function spaces then there are complex structures on it that can be extended to a complex structure of the twisted Hilbert space. Regarding the hyperplane problem we show that no complex structure on $\ell_2$ can be extended to a complex structure on an hyperplane of $Z_2$ containing it.