no. 5
When Kalton and Peck met Fourier
Cabello Sánchez, Félix1; Salguero-Alarcón, Alberto1
1 Instituto de Matemáticas Universidad de Extremadura Avenida de Elvas 06071-Badajoz (Spain)
Annales de l'Institut Fourier, Volume 73 (2023) no. 5, pp. 1843-1883.

The paper studies short exact sequences of Banach modules over the convolution algebra L 1 =L 1 (G), where G is a compact abelian group. The main tool is the notion of a nonlinear centralizer, which in combination with the Fourier transform, is used to produce sequences of L 1 -modules 0→L q →Z→L p →0 that are nontrivial as long as the general theory allows it, namely for p∈(1,∞],q∈[1,∞). Concrete examples are worked in detail for the circle group, with applications to the Hardy classes, and the Cantor group.

L’article étudie des suites exactes courtes de modules de Banach sur l’algèbre de convolution L 1 =L 1 (G), où G est un groupe abélien compact. L’outil principal est la notion de centralisateur non linéaire, qui, en combinaison avec la transformée de Fourier, est utilisée pour produire des suites exactes de L 1 -modules de la forme 0→L q →Z→L p →0 qui sont non triviales tant que la théorie générale le permet, à savoir pour p∈(1,∞],q∈[1,∞). Des exemples concrets sont détaillés pour le groupe du cercle, avec des applications aux classes de Hardy, et le groupe de Cantor.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3562
Classification: 43A15, 43A20, 43A25, 46M18
Keywords: Short exact sequence, Banach module, Convolution algebra, Fourier transform, Nonlinear centralizer, Riesz product.
Mots-clés : Suite exacte courte, Module de Banach, Algèbre de convolution, Transformée de Fourier, Centralisateur non linéaire, Produit de Riesz.

Cabello Sánchez, Félix 1; Salguero-Alarcón, Alberto 1

1 Instituto de Matemáticas Universidad de Extremadura Avenida de Elvas 06071-Badajoz (Spain)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Cabello Sánchez, Félix; Salguero-Alarcón, Alberto. When Kalton and Peck met Fourier. Annales de l'Institut Fourier, Volume 73 (2023) no. 5, pp. 1843-1883. doi : 10.5802/aif.3562. https://aif.centre-mersenne.org/articles/10.5802/aif.3562/

[1] Brudnyi, Yuri A.; Kalton, Nigel J. Polynomial approximation on convex subsets of ℝ n , Constr. Approx., Volume 16 (2000) no. 2, pp. 161-199 | DOI | MR | Zbl

[2] Cabello Sánchez, Félix Nonlinear centralizers in homology, Math. Ann., Volume 358 (2014) no. 3-4, pp. 779-798 | DOI | MR | Zbl

[3] Cabello Sánchez, Félix; Castillo, Jesús M. F. Uniform boundedness and twisted sums of Banach spaces, Houston J. Math., Volume 30 (2004) no. 2, pp. 523-536 | MR | Zbl

[4] Cabello Sánchez, Félix; Castillo, Jesús M. F. Homological Methods in Banach Space Theory, Cambridge Studies in Advanced Mathematics, 203, Cambridge University Press, 2023 | DOI

[5] Cabello Sánchez, Félix; Castillo, Jesús M. F.; Kalton, Nigel J.; Yost, David T. Twisted sums with C(K) spaces, Trans. Am. Math. Soc., Volume 355 (2003) no. 11, pp. 4523-4541 | MR | Zbl

[6] Cabello Sánchez, Félix; Castillo, Jesús M. F.; Salguero-Alarcón, Alberto The behaviour of quasi-linear maps on C(K)-spaces, J. Math. Anal. Appl., Volume 475 (2019) no. 2, pp. 1714-1719 | DOI | MR | Zbl

[7] Cabello Sánchez, Félix; Castillo, Jesús M. F.; Suárez, Jesús On strictly singular nonlinear centralizers, Nonlinear Anal., Theory Methods Appl., Volume 75 (2012) no. 7, pp. 3313-3321 | MR | Zbl

[8] Castillo, Jesús M. F.; Ferenczi, Valentin Group actions on twisted sums of Banach spaces (2020) (https://arxiv.org/abs/2003.09767)

[9] Castillo, Jesús M. F.; Ferenczi, Valentin; González, Manuel Singular twisted sums generated by complex interpolation, Trans. Am. Math. Soc., Volume 369 (2017) no. 7, pp. 4671-4708 | DOI | MR | Zbl

[10] Castillo, Jesús M. F.; Simões, Marilda A. Positions in ℓ 1 , Banach J. Math. Anal., Volume 9 (2015) no. 4, pp. 395-404 | DOI | Zbl

[11] Cohn, Donald L. Measure Theory, Birkhäuser Advanced Texts, Birkhäuser, 2013 | DOI | MR

[12] Dales, Harold G. Banach algebras and automatic continuity, London Mathematical Society Monographs, 24, Clarendon Press, 2000

[13] Diestel, Joe; Jarchow, Hans; Tonge, Andrew Absolutely summing operators, Cambridge Studies in Advanced Mathematics, 43, Cambridge University Press, 1995 | DOI

[14] Duren, Peter L. Theory of H p spaces, Academic Press Inc., 1970

[15] Gardner, Martin Mathematical Circus, The Mathematical Association of America, 1992

[16] Grafakos, Loukas Classical Fourier Analysis, Graduate Texts in Mathematics, 249, Springer, 2014 | DOI

[17] Greenleaf, Frederick P. Invariant Means on Topological Groups, 16, Van Nostrand Reinhold Company, 1969

[18] Haagerup, Uffe The best constants in the Khintchine inequality, Stud. Math., Volume 70 (1981) no. 3, pp. 231-283 | DOI | MR

[19] Hewitt, Edwin; Ross, Kenneth A. Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact abelian groups, 1970, Springer, 1970

[20] Kalton, Nigel J. The three-space problem for locally bounded F-spaces, Compos. Math., Volume 37 (1978), pp. 243-276 | Numdam | MR | Zbl

[21] Kalton, Nigel J. Nonlinear commutators in interpolation theory, Memoirs of the American Mathematical Society, 73, American Mathematical Society, 1988

[22] Kalton, Nigel J.; Peck, N. Tenney Twisted sums of sequence spaces and the three-space problem, Trans. Am. Math. Soc., Volume 255 (1979), pp. 1-30 | DOI | MR | Zbl

[23] Kalton, Nigel J.; Roberts, James W. Uniformly exhaustive submeasures and nearly additive set functions, Trans. Am. Math. Soc., Volume 278 (1983), pp. 803-816 | DOI | MR | Zbl

[24] Katznelson, Yitzhak An Introduction to Harmonic Analysis, Cambridge Mathematical Library, Cambridge University Press, 2004 | DOI

[25] Kochanek, Tomasz Stability of vector measures and twisted sums of Banach spaces, J. Funct. Anal., Volume 264 (2013) no. 10, pp. 2416-2456 | DOI | MR | Zbl

[26] Kolwicz, Paweł; Leśnik, Karol; Maligranda, Lech Pointwise products of some Banach function spaces and factorization, J. Funct. Anal., Volume 266 (2014) no. 2, pp. 616-659 | DOI | MR | Zbl

[27] Kuchment, Peter A. Three-representation problem in Banach spaces, Complex Anal. Oper. Theory, Volume 15 (2021) no. 2, 34, 23 pages | MR

[28] Lindenstrauss, Joram On a certain subspace of ℓ 1 , Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys., Volume 12 (1964), pp. 539-542 | Zbl

[29] Lopez, Jorge M.; Ross, Kenneth A. Sidon Sets, Lecture Notes in Pure and Applied Mathematics, 13, Marcel Dekker, 1975

[30] Lusky, Wolfgang A note on Banach spaces containing c 0 or C ∞ , J. Funct. Anal., Volume 62 (1985), pp. 1-7 | DOI | MR | Zbl

[31] Pisier, Gilles Les inégalités de Khintchine-Kahane d’après C. Borell, Seminaire sur la géométrie des espaces de Banach. 1977-1978, École Polytech., 1978, pp. 1977-1978 (Exp. No. 7, 14 pp.) | Numdam | Zbl

[32] Pisier, Gilles Grothendieck’s Theorem, past and present, Bull. Am. Math. Soc., Volume 49 (2012) no. 2, pp. 237-323 | DOI | MR | Zbl

[33] Rudin, Walter Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, 12, Interscience Publishers, division of John Wiley and Sons, 1962

[34] Runde, Volker Amenable Banach Algebras. A Panorama, Springer Monographs in Mathematics, Springer, 2020 | DOI

[35] Wojtaszczyk, Przemyslaw Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, 25, Cambridge University Press, 1991 | DOI

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