@article{an Huef_Laca_Raeburn_2021, title={Boundary quotients of the right Toeplitz algebra of the affine semigroup over the natural numbers}, volume={52}, url={https://nzjmath.org/index.php/NZJMATH/article/view/90}, DOI={10.53733/90}, abstractNote={<p>We study the Toeplitz $C^*$-algebra generated by the right-regular representation of the semigroup ${\mathbb N \rtimes \mathbb N^\times}$, which we call the right Toeplitz algebra. We analyse its structure by studying three distinguished quotients. We show that the multiplicative boundary quotient is isomorphic to a crossed product of the Toeplitz algebra of the additive rationals by an action of the multiplicative rationals, and study its ideal structure. The Crisp--Laca boundary quotient is isomorphic to the $C^*$-algebra of the group ${\mathbb Q_+^\times}\!\! \ltimes \mathbb Q$. There is a natural dynamics on the right Toeplitz algebra and all its KMS states factor through the additive boundary quotient. We describe the KMS simplex for inverse temperatures greater than one.</p>}, journal={New Zealand Journal of Mathematics}, author={an Huef, Astrid and Laca, Marcelo and Raeburn, Iain}, year={2021}, month={Sep.}, pages={109–143} }