A $c_0$ saturated Banach space with tight structure

It is shown that variants of the HI methods could yield objects closely connected to the classical Banach spaces. Thus we present a new $c_0$ saturated space, denoted as ${\mathfrak{X}}_0$, with rather tight structure. The space ${\mathfrak{X}}_0$ is not embedded into a space with an unconditional basis and its complemented subspaces have the following structure. Everyone is either of type I, namely, contains an isomorph of ${\mathfrak{X}}_0$ itself or else is isomorphic to a subspace of $c_0$ (type II). Furthermore for any analytic decomposition of ${\mathfrak{X}}_0$ into two subspaces one is of type I and the other is of type II. The operators of ${\mathfrak{X}}_0$ share common features with those of HI spaces.