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# Multi angle

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Multi angle The story of Kalton's last unpublished paper

Auteurs : Castillo, Jesús M.F. (Auteur de la Conférence)
CIRM (Editeur )

Résumé : I'd like to share with the audience the Kaltonian story behind [1], started in 2004, including the problems we wanted to solve, and could not.
In that paper we show that Rochberg's generalized interpolation spaces $\mathbb{Z}^{(n)}$ [5] can be arranged to form exact sequences $0\to\mathbb{Z}^{(n)}\to\mathbb{Z}^{(n+k)}\to\mathbb{Z}^{(k)} \to 0$. In the particular case of Hilbert spaces obtained from the interpolation scale of $\ell_p$ spaces then $\mathbb{Z}^{(2)}$ becomes the well-known Kalton-Peck $Z_2$ space, and one gets from here that there are quite natural nontrivial twisted sums $0\to Z_2\to\mathbb{Z}^{(4)}\to Z_2 \to0$ of $Z_2$ with itself. The twisted sum space $\mathbb{Z}^{(4)}$ does not embeds in, or is a quotient of, a twisted Hilbert space and does not contain $\ell_2$ complemented. We will also construct another nontrivial twisted sum of $Z_2$ with itself that contains $\ell_2$ complemented. These results have some connection with the nowadays called Kalton calculus [3, 4], and thus several recent advances [2] in this theory that combines twisted sums and interpolation theory will be shown.

Banach space - twisted sum - complex interpolation - Hilbert space

Codes MSC :
46B20 - Geometry and structure of normed linear spaces
46M18 - Homological methods

 Informations sur la Vidéo Réalisateur : Hennenfent, Guillaume Langue : Anglais Date de publication : 21/01/15 Date de captation : 14/01/15 Collection : Research talks Format : quicktime ; audio/x-aac Durée : 00:46:27 Domaine : Analyse & Applications Audience : Chercheurs ; Doctorants , Post - Doctorants Download : http://videos.cirm-math.fr/2015-01-14_Castillo.mp4 Informations sur la rencontre Nom du congrès : Banach spaces and their applications in analysis / Espaces de Banach et applications à l'analyseOrganisteurs Congrès : Albiac, Fernando ; Casazza, Peter G. ; Godefroy, GillesDates : 12/01/15 - 16/01/15 Année de la rencontre : 2015 Citation DataDOI : 10.24350/CIRM.V.18666103Cite this video as: Castillo, Jesús M.F. (2015). The story of Kalton's last unpublished paper. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18666103URI : http://dx.doi.org/10.24350/CIRM.V.18666103

Bibliographie

1. [1] Cabello, F., Castillo, J.M.F., & Kalton, N.J. (2014). Complex interpolation and twisted twisted Hilbert spaces. <arXiv:1406.6723> - http://arxiv.org/abs/1406.6723v1

2. [2] Castillo, J.M.F., Ferenczi, V., & Gonzalez, M. (2014). Singular twisted sums generated by complex interpolation. <arXiv:1410.5505> - http://arxiv.org/abs/1410.5505v1

3. [3] Kalton, N.J. (1988). Nonlinear commutators in interpolation theory. Memoirs of the American Mathematical Society, 385, 85 p. - https://zbmath.org/?q=an:0658.46059

4. [4] Kalton, N.J. (1992). Differentials of complex interpolation processes for Köthe function spaces. Transactions of the American Mathematical Society, 333(2), 479-529 - http://dx.doi.org/10.1090/s0002-9947-1992-1081938-1

5. [5] Rochberg, R. (1996). Higher order estimates in complex interpolation theory. Pacific Journal of Mathematics, 174(1), 247-267 - https://www.zbmath.org/?q=an:0866.46047

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