An Ando-type dilation on right LCM monoids

TL;DR

This work extends Ando-type dilation results to the setting of a pair of commuting contractions together with a representation of a countable right LCM monoid $P$ realized via either the Cartesian or the free product. By leveraging $*$-regular dilation theory on right-LCM monoids and a Davidson–Katsoulis strategy, the authors prove that if each $S_i\times T$ admits a $*$-regular dilation, then $(S_1\times S_2)\times T$ admits an isometric dilation to commuting isometries with an isometric representation of $P$ (and similarly for the free product). The results generalize Barik–Das from $\mathbb{N}^k$ to arbitrary right-LCM monoids and extend to the free product setting, yielding new Ando-type dilations for $\mathbb{N}^2\times P$ and $\mathbb{N}^2*P$, with further implications for right-angled Artin monoids under a graph-theoretic condition. The paper also discusses the connections to Frazho–Bunce–Popescu dilations, double commutativity, and the limitations on obtaining stronger Nica-covariant dilations in general.

Abstract

We establish an Ando-type dilation theorem for a pair of commuting contractions together with a representation of a right LCM monoid via either the Cartesian or the free product. We prove that if each individual contraction together with the monoid representation has $*$-regular dilation, then they can be dilated to commuting isometries and an isometric representation of the monoid. This extends an earlier result of Barik and Das.

Theorems & Definitions (15)

• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• Theorem 3.1
• proof
• Remark 3.2
• Example 3.3
• Example 3.4
• Theorem 3.5