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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Singular twisted sums generated by complex interpolation


Authors: Jesús M. F. Castillo, Valentin Ferenczi and Manuel González
Journal: Trans. Amer. Math. Soc. 369 (2017), 4671-4708
MSC (2010): Primary 46M18, 46B70, 46E30
DOI: https://doi.org/10.1090/tran/6809
Published electronically: November 28, 2016
MathSciNet review: 3632546
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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}$.


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Additional Information

Jesús M. F. Castillo
Affiliation: Departamento de Matemáticas, Universidad de Extremadura, Avenida de Elvas s/n, 06011 Badajoz, España
Email: castillo@unex.es

Valentin Ferenczi
Affiliation: Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, rua do Matão 1010, 05508-090 São Paulo SP, Brazil – and – Equipe d’Analyse Fonctionnelle, Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie - Paris 6, Case 247, 4 place Jussieu, 75252 Paris Cedex 05, France
Email: ferenczi@ime.usp.br

Manuel González
Affiliation: Departamento de Matemáticas, Universidad de Cantabria, Avenida de los Castros s/n, 39071 Santander, España
Email: manuel.gonzalez@unican.es

DOI: https://doi.org/10.1090/tran/6809
Received by editor(s): January 16, 2015
Received by editor(s) in revised form: July 10, 2015
Published electronically: November 28, 2016
Additional Notes: This research was supported by Project MTM2013-45643, D.G.I. Spain
The research of the second author was supported by Fapesp project 2013/11390-4, including visits of the first and third authors to the University of São Paulo
Article copyright: © Copyright 2016 American Mathematical Society

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