Twist and Shout - a continuation

We consider the class of Banach spaces Y for which the space c_0 admits a nontrivial twisted sum with Y, i.e., there exists a Banach space X containing a noncomplemented copy Z of c_0 such that the quotient space X/Z is isomorphic to Y. We present a characterization of such spaces Y in terms of properties of the weak* topology of the dual space Y*. We prove that under the continuum hypothesis c_0 has a nontrivial twisted sum with every space of the form Y=C(K), where K is compact and not metrizable. This gives a consistent affirmative solution to a problem posed by Cabello, Castillo, Kalton and Yost (earlier, together with G. Plebanek, we gave a negative answer to this problem, assuming Martin Axiom and the negation of the continuum hypothesis). This is a joint research with Antonio Aviles and Grzegorz Plebanek; the preprint with these results can be found here: https://arxiv.org/abs/1902.07783 (see also https://www.youtube.com/watch?v=b-VAxGJdJeQ).