Ideal structure and factorization properties of the regular kernel operators

Abstract

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(mu ) space, with mu sigma -finite and 1<p<infty , the norm-closure of the ideal of finite-rank operators on L^p(mu ), 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(mu ) space (mu and p as before) factors with regular factors through ell _p. We show that a similar but weaker factorization property, where ell _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.