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On the bounded approximation property on subspaces of ℓp when 0 < p < 1 and related issues

  • Félix Cabello Sánchez ORCID logo EMAIL logo , Jesús M. F. Castillo and Yolanda Moreno
Published/Copyright: August 14, 2019
Forum Mathematicum
From the journal Forum Mathematicum Volume 31 Issue 6

Abstract

This paper studies the bounded approximation property (BAP) in quasi-Banach spaces. In the first part of the paper, we show that the kernel of any surjective operator ℓp→X has the BAP when X has it and 0<p≤1, which is an analogue of the corresponding result of Lusky for Banach spaces. We then obtain and study nonlocally convex versions of the Kadec–Pełczyński–Wojtaszczyk complementably universal spaces for Banach spaces with the BAP.

Keywords: Bounded approximation property; quasi-Banach space; complementably universal; Fraïssé limit
MSC 2010: 46B08; 46M07; 46B26

Communicated by Siegfried Echterhoff

Funding source: Dirección General de Investigación Científica y Técnica

Award Identifier / Grant number: MTM2016-76958-C2-1-P

Funding source: Consejería de Educación y Empleo, Junta de Extremadura

Award Identifier / Grant number: IB-16056

Funding statement: Supported in part by MINCIN project MTM2016-76958-C2-1-P (Spain) and Junta de Extremadura project IB-16056.

A Appendix: Pull-back

In the papers [3, 6, 9], “amalgamations” are invariably constructed by means of push-outs. Here we adopt the “dual view point” which is perhaps more direct since it does not depend on the ambient category nor uses quotients.

Admittedly, this sounds a bit cryptic. The point is the following. Suppose we are given two p-Banach spaces X and Y. Then its direct sum in the “isometric” category of p-Banach spaces is the space X×Y with the p-norm ∥(x,y)∥=(∥x∥p+∥y∥p)1/p. But each p-Banach space is also a q-Banach space for 0<q<p, so one could consider the q-Banach direct sum, which carries a different quasinorm. Note that, in both cases, the direct product is X×Y quasinormed with ∥(x,y)∥=max⁡(∥x∥,∥y∥). This discussion can be extrapolated to the push-out and pull-back constructions: the former is a quotient of the direct sum, while the latter is a subspace of the pull-back.

So let us have a little chat about the pull-back construction for quasi-Banach spaces. We first explain what we need and then how one can manage to get it. Let α:A→E and β:B→E be operators acting between p-Banach spaces. What we need is another p-Banach space PB, together with contractive operators α¯ and β¯ making the following square commutative:

Moreover, and this is the crucial point, the square has to be “minimally commutative” in the sense that, for every couple of operators β′:C→A and α′:C→B satisfying α⁢β′=β⁢α′, there is a unique operator γ:C→PB such that

  1. α′=α¯⁢γ and β′=β¯⁢γ and β′′=β′⁢γ,

  2. ∥γ∥≤max⁡(∥α′∥,∥β′∥).

This universal property can be visualized in the commutative diagram

(A.1)

hence the term “pull-back”.

In particular, the space PB is unique, up to isometries. Having said this, let us describe a “concrete” representation. The pull-back space is PB=PB⁢(α,β)={(a,b)∈A⊕∞B:α⁢(a)=β⁢(b)}. The arrows under bars are the restriction of the projections onto the corresponding factor. These work as required since if β′:C→A and α′:C→B satisfy α⁢β′=β⁢α′, then we can set γ⁢(c)=(β′⁢(c),α′⁢(c)).

It is important to realize that if α and β are rational maps, then PB has a basis of rational vectors of A×B=𝕂n+m and, therefore, it can be regarded as 𝕂k equipped with a p-norm that has to be allowed if those of A and B are, by conditions (2) and (3) in the definition of Section 3.3. The “projections” α¯ and β¯ are then rational maps and the universal property of diagram A.1 “preserves” rational maps in the sense that if β′ and α′ are rational maps, then so is γ.

Please keep this in mind when reading the proof of Lemma 3.3.

Acknowledgements

The authors thank Jordi López-Abad for inspiring conversations about Fraïssé limits. Because … they were inspiring, weren’t they?

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Received: 2018-07-24
Revised: 2019-03-16
Published Online: 2019-08-14
Published in Print: 2019-11-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Censored symmetric Lévy-type processes
  3. 𝐾-theory and immersions of spatial polygon spaces
  4. 𝐻𝑝 spaces for generalized Schrödinger operators and applications
  5. Commutative cocycles and stable bundles over surfaces
  6. Normal elements in the mod-𝑝 Iwasawa algebra over SL𝑛(ℤ𝑝): A computational approach
  7. Non-spectrality of self-affine measures on the three-dimensional Sierpinski gasket
  8. Plurisubharmonic functions on a neighborhood of a torus leaf of a certain class of foliations
  9. Discrete Littlewood–Paley–Stein characterization of multi-parameter local Hardy spaces
  10. Exceptional sets for sums of almost equal prime cubes
  11. Higher differentiability of solutions to a class of obstacle problems under non-standard growth conditions
  12. Beilinson–Flach elements, Stark units and 𝑝-adic iterated integrals
  13. On the bounded approximation property on subspaces of ℓp when 0 < p < 1 and related issues
  14. Refinement of the Chowla–Erdős method and linear independence of certain Lambert series
  15. Metric geometry of infinite-dimensional Lie groups and their homogeneous spaces
  16. On commutator Krylov transitive and commutator weakly transitive Abelian p-groups
https://doi.org/10.1515/forum-2018-0174
Keywords for this article
Bounded approximation property; quasi-Banach space; complementably universal; Fraïssé limit

Articles in the same Issue

  1. Frontmatter
  2. Censored symmetric Lévy-type processes
  3. 𝐾-theory and immersions of spatial polygon spaces
  4. 𝐻𝑝 spaces for generalized Schrödinger operators and applications
  5. Commutative cocycles and stable bundles over surfaces
  6. Normal elements in the mod-𝑝 Iwasawa algebra over SL𝑛(ℤ𝑝): A computational approach
  7. Non-spectrality of self-affine measures on the three-dimensional Sierpinski gasket
  8. Plurisubharmonic functions on a neighborhood of a torus leaf of a certain class of foliations
  9. Discrete Littlewood–Paley–Stein characterization of multi-parameter local Hardy spaces
  10. Exceptional sets for sums of almost equal prime cubes
  11. Higher differentiability of solutions to a class of obstacle problems under non-standard growth conditions
  12. Beilinson–Flach elements, Stark units and 𝑝-adic iterated integrals
  13. On the bounded approximation property on subspaces of ℓp when 0 < p < 1 and related issues
  14. Refinement of the Chowla–Erdős method and linear independence of certain Lambert series
  15. Metric geometry of infinite-dimensional Lie groups and their homogeneous spaces
  16. On commutator Krylov transitive and commutator weakly transitive Abelian p-groups
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