Contractive representations of odometer semigroup

TL;DR

This paper studies contractive representations of the odometer semigroup, providing a complete description of its representations on Fock space, and classifies Nica covariant representations, advancing understanding of operator representations in this context.

Contribution

It offers a complete characterization of representations of the odometer semigroup on Fock space and classifies Nica covariant representations, revealing new structural insights.

Findings

Contractive representations of $O_n$ admit nicer forms.

Complete description of $O_n$ representations on Fock space.

Classification of Nica covariant representations.

Abstract

Given a natural number $n \geq 1$, the odometer semigroup $O_n$, also known as the adding machine or the Baumslag-Solitar monoid with two generators, is a well-known object in group theory. This paper examines the odometer semigroup in relation to representations of bounded linear operators. We focus on noncommutative operators and prove that contractive representations of $O_n$ always admit to nicer representations of $O_n$. We give a complete description of representations of $O_n$ on the Fock space and relate it to the odometer lifting and subrepresentations of $O_n$. Along the way, we also classify Nica covariant representations of $O_n$.