Semilattice Structures of Spreading Models

Given a Banach space X, denote by SPw(X) the set of equivalence classes of spreading models of X generated by normalized weakly null sequences in X. It is known that SPw(X) is a semilattice, i.e., it is a partially ordered set in which every pair of elements has a least upper bound. We show that every countable semilattice that does not contain an infinite increasing sequence is order isomorphic to SPw(X) for some separable Banach space X. Given a normalized basic sequence (yi) in a Banach space and εn ↘ 0, using Ramsey’s Theorem, one can find a subsequence (xi) and a normalized basic sequence (x̃i) such that for all n ∈ N and (ai)i=1 ⊆ [−1, 1], |‖ ∑ aixki‖ − ‖ ∑ aix̃i‖| < εn for all n ≤ k1 < · · · < kn. The sequence (x̃i) is called a spreading model of (xi). It is well-known that if (xi) is in addition weakly null, then (x̃i) is 1-spreading and suppression 1-unconditional. See [3, 5] for more about spreading models. A spreading model (x̃i) is said to (C-) dominate another spreading model (ỹi) if there is a C <∞ such that for all (ai) ⊆ R, ‖ ∑ aiỹi‖ ≤ C‖ ∑ aix̃i‖. The spreading models (x̃i) and (ỹi) are said to be equivalent if they dominate each other. Let [(x̃i)] denote the class of all spreading models which are equivalent to (x̃i). Let SPw(X) denote the set of all [(x̃i)] generated by normalized weakly null sequences in X. If [(x̃i)], [(ỹi)] ∈ SPw(X), we write [(x̃i)] ≤ [(ỹi)] if (ỹi) dominates (x̃i). (SPw(X),≤) is a partially ordered set. The paper [2] initiated the study of the order structures of SPw(X). It was established that every countable subset of (SPw(X),≤) admits an upper bound ([2, Proposition 3.2]). Moreover, from the proof of this result, it follows that every pair of elements in (SPw(X),≤) has a least upper bound. In other words, (SPw(X),≤) is a semilattice. In [6], it was shown that if SPw(X) is countable, then it cannot admit a strictly increasing infinite sequence (x̃i ) < (x̃ 2 i ) < · · · . In [4], two methods of construction, utilizing Lorentz sequence spaces and Orlicz sequence spaces respectively, were used to produce Banach spaces X so that SPw(X) has certain prescribed order 2000 Mathematics Subject Classification. 46B20, 46B15.