AbstractWe consider the structure of the lattice of (order and algebra) ideals of the band of regular kernel operators on $$L^p$$
L
p
-spaces. We show, in particular, that for any $$L^p(\mu )$$
L
p
(
μ
)
space, with $$\mu $$
μ
$$\sigma $$
σ
-finite and $$1<p<\infty $$
1
<
p
<
∞
, the norm-closure of the ideal of finite-rank operators on $$L^p(\mu )$$
L
p
(
μ
)
, is the only non-trivial proper closed (order and algebra) ideal of this band. Key to our results in the $$L^p$$
L
p
setting is the fact that every regular kernel operator on an $$L^p(\mu )$$
L
p
(
μ
)
space ($$\mu $$
μ
and p as before) factors with regular factors through $$\ell _p$$
ℓ
p
. We show that a similar but weaker factorization property, where $$\ell _p$$
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.
Limits of kernel operators and the spectral regularity lemma