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

Abstract: We consider the semigroup crossed product of the additive natural numbers by the multiplicative natural numbers. We study its Toeplitz C*-algebra generated by the right-regular representation, 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. We identify the Crisp-Laca boundary quotient as the C*-algebra of the corresponding group built from rational numbers. 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.