Singular twisted sums generated by complex interpolation

TL;DR

This paper introduces new methods for constructing singular twisted sums using complex interpolation, leading to novel examples including the Kalton-Peck space and twisted sums of H.I. spaces, expanding understanding of Banach space structures.

Contribution

The paper develops general techniques for creating singular twisted sums from complex interpolation, producing the first H.I. twisted sum of an H.I. space and new twisted Hilbert spaces.

Findings

Constructed the Kalton-Peck space $Z_2$ as a singular twisted Hilbert space.

First example of an H.I. twisted sum of an H.I. space.

Established a sequence of H.I. spaces with iterated singular twisted sums.

Abstract

We present new methods to obtain singular twisted sums $X\oplus_\Omega X$ (i.e., exact sequences $0\to X\to X\oplus_\Omega X \to X\to 0$ in which the quotient map is strictly singular), in which $X$ is the interpolation space arising from a complex interpolation scheme and $\Omega$ is the induced centralizer. Although our methods are quite general, in our applications we are mainly concerned with the choice of $X$ as either a Hilbert space, or Ferenczi's uniformly convex Hereditarily Indecomposable space. In the first case, we construct new singular twisted Hilbert spaces, including the only known example so far: the Kalton-Peck space $Z_2$. In the second case we obtain the first example of an H.I. twisted sum of an H.I. space. We then use Rochberg's description of iterated twisted sums to show that there is a sequence $\mathcal F_n$ of H.I. spaces so that $\mathcal F_{m+n}$ is a singular twisted sum of $\mathcal F_m$ and $\mathcal F_n$, while for $l>n$ the direct sum $\mathcal F_n \oplus \mathcal F_{l+m}$ is a nontrivial twisted sum of $\mathcal F_l$ and $\mathcal F_{m+n}$. We also introduce and study the notion of disjoint singular twisted sum of K\"othe function spaces and construct several examples involving reflexive $p$-convex K\"othe function spaces, which include the function version of the Kalton-Peck space $Z_2$.