If $G$ is a locally compact group, then natural spaces such as $L^1(G)$ or $M(G)$ carry more structure than just that of a Banach algebra. They are also vector lattices, so that they are, in fact, Banach lattice algebras. Therefore, if they act by convolution on, say, $L^p(G)$, it is a meaningful question to ask if the corresponding map into the Banach lattice algebra $L_r(L^p(G))$ of regular operators on $L^p(G)$ is not only an algebra homomorphism, but also a lattice homomorphism. Analogous questions can be asked in similar situations, such as the left regular representation of $M(G)$.

In this lecture, we shall give an overview of what is known in this direction, and which approaches are available. The rule of thumb, based on an underlying general principle, seems to be that the answer is affirmative whenever the question is meaningful.

This is joint work with Garth Dales and David Kok.