Singular twisted sums generated by complex interpolation

Author:

Castillo Jesús,Ferenczi Valentin,González Manuel

Abstract

We present new methods to obtain singular twisted sums X Ω X X\oplus _\Omega X (i.e., exact sequences 0 X X Ω X X 0 0\to X\to X\oplus _\Omega X \to X\to 0 in which the quotient map is strictly singular) when X X is an interpolation space arising from a complex interpolation scheme and Ω \Omega is the induced centralizer.

Although our methods are quite general, we are mainly concerned with the choice of X 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 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 p -convex Köthe function spaces (which includes the function space version of the Kalton-Peck space Z 2 Z_2 ).

We then use Rochberg’s description of iterated twisted sums to show that there is a sequence F n \mathcal F_n of H.I. spaces so that F m + n \mathcal F_{m+n} is a singular twisted sum of F m \mathcal F_m and F n \mathcal F_n , while for l > n l>n the direct sum F n F l + m \mathcal F_n \oplus \mathcal F_{l+m} is a nontrivial twisted sum of F l \mathcal F_l and F m + n \mathcal F_{m+n} .

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

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