Regular Dilation and Nica-covariant Representation on Right LCM Semigroups

TL;DR

This paper extends the theory of regular dilation and Nica-covariant representations from graph products to right LCM semigroups, establishing key equivalences and conditions for *-regular dilation.

Contribution

It generalizes the equivalence between *-regular dilation and Nica-covariant dilations to right LCM semigroups, including a Brehmer-type condition.

Findings

Extended regular dilation theory to right LCM semigroups.

Established equivalence among *-regular dilation, Nica-covariant dilation, and Brehmer condition.

Applicable to various semigroups for dilation conditions.

Abstract

Regular dilation has recently been extended to graph product of $\mathbb{N}$, where having a *-regular dilation is equivalent to having a minimal isometric Nica-covariant dilation. In this paper, we first extend the result to right LCM semigroups, and establish a similar equivalence among *-regular dilation, minimal isometric Nica-covariant dilation, and a Brehmer-type condition. This result can be applied to various semigroups to establish conditions for *-regular dilation.