Semilattice Structures of Spreading Models

TL;DR

This paper characterizes the structure of spreading models in Banach spaces, showing that any countable semilattice without infinite increasing sequences can be realized as the set of spreading models of some separable Banach space.

Contribution

It establishes a correspondence between certain countable semilattices and the spreading models of separable Banach spaces, expanding understanding of their structural possibilities.

Findings

Every countable semilattice without infinite increasing sequences is realizable as SP_{w}(X).

SP_{w}(X) forms a semilattice structure in Banach spaces.

The result applies to separable Banach spaces, broadening the class of known spreading model structures.

Abstract

Given a Banach space X, denote by SP_{w}(X) the set of equivalence classes of spreading models of X generated by normalized weakly null sequences in X. It is known that SP_{w}(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 SP_{w}(X) for some separable Banach space X.