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Jesús M. F. Castillo: A few problems on twisted Hilbert spaces

**Data:** segunda-feira, 21 de outubro de 2013, às 14h
**Sala:** 267-A
**Palestrante:** Jesús M. F. Castillo, Universidad de Extremadura

**Título:** A few problems on twisted Hilbert spaces

**Resumo:** A Banach space is called a twisted Hilbert space if there is an exact sequence
$$0\longrightarrow \ell_2 \longrightarrow X \longrightarrow \ell_2 \longrightarrow 0$$
which exactly means that $X$ admits a subspace isomorphic to $\ell_2$ such that the corresponding quotient $X/\ell_2$ is isomorphic to $\ell_2$. Of course, one is interested in *nontrivial* twistings; i.e., that $X$ is not a Hilbert space. Here is a sampler of open problems involving twisted Hilbert spaces we could well consider during the talk:
- Do nontrivial twisted Hilbert spaces exist? (a question non posed by Palais). How can one construct them?
- Can they be twisted? (an old question of David Yost) i.e., Do nontrivial twisted twisted Hilbert spaces exist?
- Rochberg constructions [
*Higher order estimates in complex interpolation theory,* Pacific J. Math. 174 (1996) 247--267.] provide natural and simple representations for twisted-twisted-$\cdots$-twisted Hilbert spaces.
- The most natural example of which is the so-called Kalton-Peck $Z_2$-space and its associated exact sequence $$0 \longrightarrow \ell_2 \longrightarrow Z_2 \longrightarrow \ell_2 \longrightarrow 0.$$ Both the space $Z_2$ and the sequence are singular objects in many senses. For instance, $Z_2$ is the only natural candidate to Banach space not isomorphic to its hyperplanes.
- Homological tensorization and Kalton [
*The basic sequence problem*, Studia Math. 116 (1995) 167--187] (quasi-Banach) version of the Gowers-Maurey space [*Banach spaces with small spaces of operators,* Math. Ann. 307 (1997) 543--568] suggest that awesomer monsters are in stock.
- There is a connection between the problem of twisting Hilbert spaces and the problem of extending bilinear forms.
- In turn, the right context for such study is homological algebra. Unfortunately, the problem at the horizon (somehow a twisted reading of Palamodov [
*Homological methods in the theory of locally convex spaces,* Uspehi Mat. Nauk 26 (1971), n. 1 (157), 3--65]) **Problem**. Is $Ext^2(\ell_2, \ell_2)=0$? is entirely off limits.
- Even using Kalton extrapolation theorem [
*Differentials of complex interpolation processes for Köthe function spaces,* Trans. Amer. Math. Soc. 333 (1992) 479--529]?