Quantifier elimination in the theory of Lp(Lq) Banach lattices

We introduce the class of doubly atomless bands in Lp(Lq)-Banach lattices and show that this class is axiomatizable by positive bounded sentences in the language of Banach lattices. (Here p 6= q are fixed and in the interval 1 ≤ p, q < ∞ .) The theory of this class is complete (indeed, we show it is separably categorical) and model complete. Further, we show that it satisfies quantifier elimination if and only if the ratio p/q is not an integer. On the functional analytic side, the proof of the latter result uses a positive-coefficient version of the well known Rudin-Plotkin-Hardin extension theorems. 2000 Mathematics Subject Classification 03C65 (primary); 46B04, 46E30, 46M07 (secondary)