On c0-saturated Banach spaces

A Banach space E is c0-saturated if every closed infinite dimensional subspace of E contains an isomorph of c0. A c0-saturated Banach space with an unconditional basis which has a quotient space isomorphic to l2 is constructed. A Banach space E is c0-saturated if every closed infinite dimensional subspace of E contains an isomorph of c0. In [2] and [3], it was asked whether all quotient spaces of c0-saturated spaces having unconditional bases are also c0-saturated. In [3], Rosenthal expressed the opinion that the answer should be no. Here, we construct an example which confirms this opinion. Standard Banach space terminology, as may be found in [1], is employed. For 1 ≤ p ≤ ∞, ‖ · ‖p denotes the l-norm, and if additionally p < ∞, ‖(an)‖p,∞ = sup ann, where (an) is the decreasing rearrangement of (|an|), is the “norm” of the Lorentz space l. The cardinality of a set A is denoted by |A|. 1 Definition of the space E and simple properties Let D = {(i, j) : i, j ∈ N I , i ≥ j} and let G be the vector lattice of all functions x : D → R I having finite support. Then let B = {b = (bi) ∈ c00 : ibi ∈ N I ∪ {0} for all i, ‖bi‖2 ≤ 1}. For all b ∈ B, define xb ∈ G by xb(i, j) = { 1 1 ≤ j ≤ ibi, i ∈ N I 0 otherwise. Let U be the convex solid hull of {xb : b ∈ B}. Define a seminorm ρ on G by ρ(x) = ∥ ∥ ∥ ∥ ( 1 i i ∑