Ideal structure and factorization properties of the regular kernel operators

We consider the structure of the lattice of (order and algebra) ideals of the band of regular kernel operators on L^p-spaces. We show, in particular, that for any L^p(μ) space, with μσ-finite and 1 < p< ∞, the norm-closure of the ideal of finite-rank operators on L^p(μ) , is the only non-trivial proper closed (order and algebra) ideal of this band. Key to our results in the L^p setting is the fact that every regular kernel operator on an L^p(μ) space (μ and p as before) factors with regular factors through ℓ_p. We show that a similar but weaker factorization property, where ℓ_p is replaced by some reflexive purely atomic Banach lattice, characterizes the regular kernel operators from a reflexive Banach lattice with weak order unit to a KB-space with weak order unit.