Singular twisted sums generated by complex interpolation

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 Köthe function spaces and construct several examples involving reflexive $p$-convex Köthe function spaces, which include the function version of the Kalton-Peck space $Z_2$.