Methods in the theory of hereditarily indecomposable Banach spaces

On spaces admitting no ℓ p or c 0 spreading model

“…The proofs can be found in [5 (1) For every x ∈ S X there is a y ∈ S ZX such that Q X y = x. In particular,…”

“…In [5] it is shown that every separable Banach space either contains ℓ 1 or is a quotient of a hereditarily indecomposable space. The main theorem of this paper is a similar dichotomy for spaces that do not admit ℓ 1 spreading models.…”

A Banach space dichotomy theorem for quotients of subspaces

“…Using methods based on the definition of some notion of HI interpolation of Banach spaces, S. Argyros and V. Felouzis constructed a HI space with some quotient space isomorphic to c 0 (resp. ℓ p , 1 < p < ∞) [2]. S. Argyros and A. Tolias used deep constructions, based on what is now known as the "extension method" [1], to prove that any separable Banach space which does not contain a copy of ℓ 1 is isomorphic to the quotient space of some separable HI space [4]; and to construct a reflexive Banach space X which is HI but whose dual is saturated with unconditional basic sequences [5], therefore any quotient space of X has a further quotient with an unconditional basis.…”

“…As noticed in the introduction, HI spaces can fall in either side of the dichotomy in Theorem 6. The example of X GM is QHI, while the examples of [2] have an unconditional quotient. The dual X * uh of the reflexive space X uh of Argyros and Tolias [5] has the following quite interesting mixed property.…”

On Hereditarily Indecomposable Banach Spaces

“…Now, on one hand, Gowers and other analysts are using the "arbitrarily distortable" technique of HI spaces developed by Gowers, Maurey and other mathematicians to other mathematical topics (see, for example, [28,29,30,[31][32][33][34][35][36], [37] and [38]). On the other hand, many mathematicians have been investigating properties of HI spaces (see, [39,40,[41][42][43]), [44,45,46,47] and [48][49][50]). It has been already found that a space of HI type or the Gowers-Maurey type can be so "bad" that it has no subspace of infinite dimensions with a separable dual [14], and can be so "nice" that it has an equivalent uniformly convex norm [42].…”