Almost Limited Sets in Banach Lattices
Jin Xi Chen, Zi Li Chen, and Guo Xing Ji

TL;DR
This paper introduces almost limited sets in Banach lattices, characterizes when they coincide with L-weakly compact sets via order continuity, and explores operator implications in $\sigma$-Dedekind complete lattices.
Contribution
It defines almost limited sets, establishes their equivalence with L-weakly compact sets under order continuity, and characterizes operators related to these sets in $\sigma$-Dedekind Banach lattices.
Findings
Almost limited sets are characterized in Banach lattices.
Order continuity of the norm is equivalent to the coincidence of almost limited and L-weakly compact sets.
Relatively weakly compact sets are almost limited if and only if all operators into c0 are almost Dunford-Pettis.
Abstract
We introduce and study the class of almost limited sets in Banach lattices, that is, sets on which every disjoint weak null sequence of functionals converges uniformly to zero. It is established that a Banach lattice has order continuous norm if and only if almost limited sets and -weakly compact sets coincide. In particular, in terms of almost Dunford-Pettis operators into , we give an operator characterization of those -Dedekind complete Banach lattices whose relatively weakly compact sets are almost limited, that is, for a -Dedekind Banach lattice , every relatively weakly compact set in is almost limited if and only if every continuous linear operator is an almost Dunford-Pettis operator.
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