On the classes of hereditarily ℓp Banach spaces

Let X denote a specific space of the class of X _α,p Banach sequence spaces which were constructed by Hagler and the first named author as classes of hereditarily ℓ_p Banach spaces. We show that for p > 1 the Banach space X contains asymptotically isometric copies of ℓ_p. It is known that any member of the class is a dual space. We show that the predual of X contains isometric copies of ℓ_p where 1/p + 1/q = 1. For p = 1 it is known that the predual of the Banach space X contains asymptotically isometric copies of c ₀. Here we give a direct proof of the known result that X contains asymptotically isometric copies of ℓ₁.