On the structure of the set of higher order spreading models
Volume 223 / 2014
Studia Mathematica 223 (2014), 149-173
MSC: 46B06, 46B25, 46B45.
DOI: 10.4064/sm223-2-3
Abstract
We generalize some results concerning the classical notion of a spreading model to spreading models of order $\xi $. Among other results, we prove that the set $SM_\xi ^w(X)$ of $\xi $-order spreading models of a Banach space $X$ generated by subordinated weakly null $\mathcal {F}$-sequences endowed with the pre-partial order of domination is a semilattice. Moreover, if $SM_\xi ^w(X)$ contains an increasing sequence of length $\omega $ then it contains an increasing sequence of length $\omega _1$. Finally, if $SM_\xi ^w(X)$ is uncountable, then it contains an antichain of size continuum.
