Totally bounded sets in the absolute weak topology

In this paper, almost Dunford-Pettis operators with ranges in c₀ are used to identify totally bounded sets in the absolute weak topology. That is, a bounded subset A of a Banach lattice E is |σ|(E,E⁾-totally bounded if and only if T(A)⊂ c₀ is relatively compact for every almost Dunford-Pettis operator T:E→c₀. As an application, we show that for two Banach lattices E and F every positive operator from E to F dominated by a PL-compact operator is PL-compact if and only if either the norm of E\, is order continuous or every order interval in F is |σ|(F,F⁾-totally bounded.