Singular twisted sums generated by complex interpolation
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- by Jesús M. F. Castillo, Valentin Ferenczi and Manuel González PDF
- Trans. Amer. Math. Soc. 369 (2017), 4671-4708 Request permission
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
- MR Author ID: 247518
- ORCID: 0000-0003-3032-966X
- 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
- MR Author ID: 360353
- ORCID: 0000-0001-5239-111X
- Email: ferenczi@ime.usp.br
- Manuel González
- Affiliation: Departamento de Matemáticas, Universidad de Cantabria, Avenida de los Castros s/n, 39071 Santander, España
- MR Author ID: 219505
- Email: manuel.gonzalez@unican.es
- 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 - © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 4671-4708
- MSC (2010): Primary 46M18, 46B70, 46E30
- DOI: https://doi.org/10.1090/tran/6809
- MathSciNet review: 3632546