Boundary quotients of the Toeplitz algebra of the affine semigroup over the natural numbers

We study the Toeplitz algebra $\TT(\N\rtimes\N^\times)$ and three quotients of this algebra: the $C^*$-algebra $\qn$ recntly introduced by Cuntz, and two new ones, which we call the additive and multiplicative boundary quotients. These quotients are universal for Nica-covariant representations of $\N\rtimes\N^\times$ satisfying extra relations, and can be realised as partial crossed products. We use the structure theory for partial crossed products to prove a uniqueness theorem for the additive boundary quotient, and use the recent analysis of KMS states on $\TT(\nxnx)$ to describe the KMS states on the two quotients. We then show that $\TT(\nxnx)$, $\qn$ and our new quotients are all interesting new examples for Larsen's theory of Exel crossed products by semigroups.