A c0 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 c0 saturated space, denoted as X0, with rather tight structure. The space X0 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 X0 itself or else is isomorphic to a subspace of c0 (type II). Furthermore for any analytic decomposition of X0 into two subspaces one is of type I and the other is of type II. The operators of X0 share common features with those of HI spaces.