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

Data: segunda-feira, 21 de outubro de 2013, às 14h

Sala: 267-A

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:
1. Do nontrivial twisted Hilbert spaces exist? (a question non posed by Palais). How can one construct them?
2. Can they be twisted? (an old question of David Yost) i.e., Do nontrivial twisted twisted Hilbert spaces exist?
3. 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.
4. 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.
5. 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.
6. There is a connection between the problem of twisting Hilbert spaces and the problem of extending bilinear forms.
7. 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.
8. Even using Kalton extrapolation theorem [Differentials of complex interpolation processes for Köthe function spaces, Trans. Amer. Math. Soc. 333 (1992) 479--529]?