K)-space is not a three-space propert

In the setting of Banach spaces, a property P is said to be a three-space property if whenever a Banach space X has a subspace Y so that both Y and the quotient space X/Y satisfy P, then X also satisfies P. It has been known for some time that ``to be isomorphic to a space of continuous functions C(K)'' is not a three-space property. In this talk we construct a remarkable example of such a fact: a Banach space X which is not isomorphic to any C(K), but it contains a copy of $c_0$ so that the quotient space $X/c_0$ is isomorphic to $c_0(\mathfrak c)$.
This is a joint work with Grzegorz Plebanek.