Regular Representations of Lattice Ordered Semigroups

TL;DR

This paper characterizes when representations of lattice ordered semigroups are regular, extending previous results and showing that certain covariant representations have regular minimal dilations.

Contribution

It provides a necessary and sufficient condition for regularity of lattice ordered semigroup representations, generalizing Brehmer's theorem and introducing analogs of commuting row contractions.

Findings

Contractive Nica-covariant representations are regular.

Minimal isometric dilations of these representations are also Nica-covariant.

Introduces an analog of commuting row contractions on lattice ordered groups.

Abstract

We establish a necessary and sufficient condition for a representation of a lattice ordered semigroup to be regular, in the sense that certain extensions are completely positive definite. This result generalizes a theorem due to Brehmer where the lattice ordered group was taken to be $\mathbb{Z}_+^\Omega$. As an immediate consequence, we prove that contractive Nica-covariant representations on lattice ordered semigroups are regular, and therefore, its minimal isometric dilation is also Nica-covariant. We also introduce an analog of commuting row contractions on lattice ordered group and show that such a representation is regular.