Weakly Lindelof determined Banach spaces not containing ℓ1(N)\ell^1(N)

The class of countably intersected families of sets is defined. For any such family we define a Banach space not containing \ell^{1}(\NN ). Thus we obtain counterexamples to certain questions related to the heredity problem for W.C.G. Banach spaces. Among them we give a subspace of a W.C.G. Banach space not containing \ell^{1}(\NN ) and not being itself a W.C.G. space

Examples of asymptotically \ell_^1 Banach spaces

Two examples of asymptotic $\ell_{1}$ Banach spaces are given. The first, $X_{u}$, has an unconditional basis and is arbitrarily distortable. The second, $X$, does not contain any unconditional basic sequence. Both are spaces of the type of Tsirelson. We thus answer a question raised by W.T.Gowers

Examples of k-iterated spreading models

It is shown that for every $k\in\mathbb{N}$ and every spreading sequence $\{e_n\}_{n\in\mathbb{N}}$ that generates a uniformly convex Banach space $E$, there exists a uniformly convex Banach space $X_{k+1}$ admitting $\{e_n\}_{n\in\mathbb{N}}$ as a $k+1$-iterated spreading model, but not as a $k$-iterated one.Comment: 16 pages, no figure

Interpolating hereditarily indecomposable Banach spaces

It is shown that every Banach space either contains $\ell ^1$ or it has an infinite dimensional closed subspace which is a quotient of a H.I. Banach space.Further on, $L^p(\lambda )$, $1<p<\infty$, is a quotient of a H.I Banach space