This is an announcement for the paper “Extension operators and twisted sums of $c_0$ and $C(K)$ spaces” by Witold Marciszewskihttps://arxiv.org/find/math/1/au:+Marciszewski_W/0/1/0/all/0/1, Grzegorz Plebanekhttps://arxiv.org/find/math/1/au:+Plebanek_G/0/1/0/all/0/1.
Abstract: We investigate the following problem posed by Cabello Sanch'ez, Castillo, Kalton, and Yost: Let $K$ be a nonmetrizable compact space. Does there exist a nontrivial twisted sum of $c_0$ and $C(K)$, i.e., does there exist a Banach space $X$ containing a non-complemented copy $Z$ of $c_0$ such that the quotient space $X/Z$ is isomorphic to $C(K)$? Using additional set-theoretic assumptions we give the first examples of compact spaces $K$ providing a negative answer to this question. We show that under Martin's axiom and the negation of the continuum hypothesis, if either $K$ is the Cantor cube $2^{\omega_1}$ or $K$ is a separable scattered compact space of height 3 and weight $\omega_1$, then every twisted sum of $c_0$ and $C(K)$ is trivial. We also construct nontrivial twisted sums of $c_0$ and $C(K)$ for $K$ belonging to several classes of compacta. Our main tool is an investigation of pairs of compact spaces $K\sebset L$which do not admit an extension operator $C(K)\rightarrow C(L)$.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1703.02139