Contractive representations of odometer semigroup

Abstract

Given a natural number, the odometer semigroup—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 it always admit nicer representations. We provide a complete description of representations on the Fock space and relate them to odometer lifting and subrepresentations. Along the way, we also classify Nica covariant representations.