Sala: 267-A
Palestrante: Michael A. Rincon V., IME-USP
Título: Optimal extensions of Holsztyński theorem for $C_0(K, X)$ spaces
Michael A. Rincon V.: Optimal extensions of Holsztyński theorem for $C_{0}(K, X)$ spaces
Data: segunda-feira, 4 de novembro de 2013, às 14h
Sala: 267-A
Palestrante: Michael A. Rincon V., IME-USP
Sala: 267-A
Palestrante: Michael A. Rincon V., IME-USP
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:
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]?
Jesús M. F. Castillo: A few problems on twisted $C(K)$-spaces
Data: segunda-feira, 14 de outubro de 2013, às 14h
Sala: 267-A
Palestrante: Jesús M. F. Castillo, Universidad de Extremadura
Título: A few problems on twisted $C(K)$-spaces
Sala: 267-A
Palestrante: Jesús M. F. Castillo, Universidad de Extremadura
Título: A few problems on twisted $C(K)$-spaces
Resumo: A Banach space X is called a twisted-$C(K)$ if there are two Banach spaces of continuous functions on some compact spaces, say $C(S)$ and $C(T)$, forming an exact sequence
$$0 \longrightarrow C(S) \longrightarrow X \longrightarrow C(T) \longrightarrow 0$$
which exactly means that $X$ admits a subspace isomorphic to $C(S)$ such that the corresponding
quotient $X/C(S)$ is isomorphic to $C(T)$. Of course, one is interested in nontrivial twistings; i.e.,
that $X$ is not $C(S) \oplus C(T)$. Here is a sampler of open problems involving twisted sums of $C(K)$-
spaces we could well consider during the talk:
$$0 \longrightarrow C(S) \longrightarrow X \longrightarrow C(T) \longrightarrow 0$$
which exactly means that $X$ admits a subspace isomorphic to $C(S)$ such that the corresponding
quotient $X/C(S)$ is isomorphic to $C(T)$. Of course, one is interested in nontrivial twistings; i.e.,
that $X$ is not $C(S) \oplus C(T)$. Here is a sampler of open problems involving twisted sums of $C(K)$-
spaces we could well consider during the talk:
- Given a nontrivial twisting $X$ of $C(S)$ and $C(K)$, it is not known a way to "paste" $S$ and $T$
in a compact $K(S, T)$ so that $X$ becomes isomorphic to $C(K(S, T))$. When possible. - It is not known if for every nonmetrizable compact $K$, there is a (nontrivial) representation $C(K) \simeq X/c_0$.
- If $\mathbb{N}^*$ denotes the compact space $\beta \mathbb{N} \setminus \mathbb{N}$, every twisting of $C(\mathbb{N}^*)$ is trivial [Avilés, Cabello, Castillo, González, Moreno, On separably injective Banach spaces. Adv. in Math. 234 (2013) 192-216]. It is not known which other $C(K)$ enjoy the same property, apart from $c_0$ or the injective spaces.
- The local versions of $C(K)$ spaces are the so-called $\mathcal L_\infty$-spaces. The twisting of $\mathcal L_\infty$-spaces conceals many surprises.
- The uniform classification of (twisted) $\mathcal L_\infty$-spaces is a tough topic. Aharoni and Lindenstrauss [Uniform equivalence between Banach spaces. Bull. Am. Math. Soc. 84 (1978) 281-283] gave an example of two non-isomorphic non-separable $\mathcal L_\infty$-spaces which are uniformly homeomorphic. It is possible to give a separable example, which solves a question in the next mentioned paper of Johnson, Lindenstrauss and Schechtman.
- A $C(K)$ space is uniformly homeomorphic to $c_0$ if and only if it is isomorphic to $c_0$ [Johnson, Lindenstrauss, Schechtman, Banach spaces determined by their uniform structures. Geom. Funct. Anal. 6, (1996) 430-470]. It is not known if "the same" occurs to twisted $C(K)$-spaces.
- $C[0, 1]$ and the Gurarii space are essentially different $\mathcal L_\infty$-spaces, in the sense that they have no common ultrapowers [Avilés, Cabello, Castillo, González, Moreno, On ultra-products of Banach space of type $\mathcal L_\infty$. Fundamenta Math. (to appear)]. It is not known how many "essentially different" $\mathcal L_\infty$-spaces are there, which is known as the Henson-Moore classification problem [Nonstandard hulls of the classical Banach spaces. Duke Math. J. 41 (1974) 277.284.]
Pierluigi Benevieri: O problema da aproximação de mapas compactas em espaços de Banach e aplicações ao grau topológico
Data: segunda-feira, 7 de outubro de 2013, às 14h
Sala: 267-A
Palestrante: Pierluigi Benevieri, IME-USP
Título: O problema da aproximação de mapas compactas em espaços de Banach e aplicações ao grau topológico
De fato, podemos provar que um resultado de aproximação existe também no caso em que o domínio da mapa é contido em uma subvariedade de dimensão finita de $E$.
Este fato permite provar resultados da teoria do grau topológico em dimensão infinita.
Sala: 267-A
Palestrante: Pierluigi Benevieri, IME-USP
Título: O problema da aproximação de mapas compactas em espaços de Banach e aplicações ao grau topológico
Resumo: Pelo que se sabe, não existe um resultado geral de aproximação de mapas completamente contínuas entre espaços de Banach por mapas $C^1$, se estamos em dimensão infinita. Sabemos muito bem que uma função real contínua, definida em um subconjunto compacto de $R^n$ pode ser aproximada, na norma sup, por uma mapa suave definida em tudo $R^n$. Um resultado análogo não existe (pelo que se sabe) se $R^n$ é substituido por um espaço de Banach $E$ de dimensão infinita, a menos que o domínio (compacto) da mapa seja contido em um subespaço de dimensão finita de $E$.
De fato, podemos provar que um resultado de aproximação existe também no caso em que o domínio da mapa é contido em uma subvariedade de dimensão finita de $E$.
Este fato permite provar resultados da teoria do grau topológico em dimensão infinita.
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