Some results on the lattice of closed ideals of L^r(X) for X of the form (oplus_i l_p^i)_q

We study the lattice of closed (order and algebra) ideals of Lr(X) when X is a Banach lattice of the form (⨁iℓip)q (p∈[1,∞], q∈[1,∞)∪{0}&p≠q). We show that for every such X, Lr(X) has a unique maximal (order and algebra) ideal. For 1<p<∞ and q∈{0,1}, we show, in particular, that the lattice of closed (order and algebra) ideals of Lr(X) contains at least five distinct ideals.