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The story of Kalton's last unpublished paper

De Jesús M.F. Castillo

Apparaît également dans la collection : Banach spaces and their applications in analysis / Espaces de Banach et applications à l'analyse

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

Informations sur la vidéo

Données de citation

  • DOI 10.24350/CIRM.V.18666103
  • Citer cette vidéo Castillo, Jesús M.F. (14/01/2015). The story of Kalton's last unpublished paper. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.18666103
  • URL https://dx.doi.org/10.24350/CIRM.V.18666103

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