Contractive representations of odometer semigroup

TL;DR

The paper develops a comprehensive operator-theoretic framework for the odometer semigroup $O_n$ by modeling representations via Fock space constructions and odometer maps $W_L$ determined by symbols $L$. It provides a complete classification of Fock representations, a precise description of isometric and Nica covariant Fock representations, and a dilation (odometer lifting) theory that embeds general contractive representations into Fock-model dilations. Invariant subspaces are analyzed through a noncommutative Beurling-Lax-Halmos approach, showing that subrepresentations of Fock representations are themselves Fock representations. The work connects dilation theory with semigroup representations, yielding explicit symbolic criteria and concrete examples that illustrate when odometer maps are unitary and how spectra behave in these settings.

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 $O_n$.

Theorems & Definitions (52)

• Definition 1.1
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