The Kalton-Peck space is the complexification of the real Kalton-Peck space

TL;DR

This paper proves that the complex Kalton-Peck space is the complexification of its real counterpart, which is constructed via real interpolation and shares key properties such as duality and singularity.

Contribution

It establishes that the complex Kalton-Peck space is the complexification of the real space derived from real interpolation, clarifying their relationship.

Findings

$Z_2$ is the complexification of $Z_2^{real}$.

$Z_2^{real}$ shares properties like isomorphism to its dual and singularity.

$Z_2^{real}$ contains no complemented copies of $ ext{ell}_2$.

Abstract

The Kalton-Peck $Z_2$ space is the derived space obtained from the scale of $\ell_p$ spaces by complex interpolation at $1/2$. If we denote by by $Z_2^{real}$ the derived space obtained from the scale of $\ell_p$ spaces by real interpolation at $(1/2, 1/2)$, we show that $Z_2$ is the complexification of $Z_2^{real}$. We also show that $Z_2^{real}$ shares the most important properties of $Z_2$: it is isomorphic to its dual, it is singular and contains no complemented copies of $\ell_2$.