### 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

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:

1. 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.
2. It is not known if for every nonmetrizable compact $K$, there is a (nontrivial) representation $C(K) \simeq X/c_0$.
3. 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.
4. 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.
5. 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.
6. 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.
7. $C[0, 1]$ 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.]