A 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 X 0 , with rather tight structure. The space 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 X 0 itself or else is isomorphic to a subspace of c 0 (type II). Furthermore for any analytic decomposition of X 0 into two subspaces one is of type I and the other is of type II. The operators of X 0 share common features with those of HI spaces.