A twisted sum of $C(K)$-spaces not isomorphic to any $C(K)$-space

A twisted sum of two Banach spaces X and Y is another space Z containing Y as a closed subspace so that Z/Y = X. In this talk we focus on twisted sums of spaces of continuous functions on compact spaces. It has been known for some time that a twisted sum of two C(K)-spaces does not need to be isomorphic to a C(K)-space (see for example [1, theorem 3.5.b] or [2]). We will focus on one recent and singular construction which serves as an example: a twisted sum of c0 and c0 (c) which is not isomorphic to any C(K)-space. This is part of a joint work with Grzegorz Plebanek.