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) when $X$ is an interpolation space arising from a complex interpolation scheme and $\Omega$ is the induced centralizer.

Although our methods are quite general, 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 (which includes 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.

During our study of singularity we introduce the notion of a disjointly singular twisted sum of Köthe function spaces and construct several examples involving reflexive $p$-convex Köthe function spaces (which includes the function space version of the Kalton-Peck space $Z_2$).

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 $\mathcal F_{m+n}$.