**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

**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:

- 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.]

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