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

TL;DR

This paper investigates the structure of the right Toeplitz algebra associated with the semigroup of natural numbers under addition and multiplication, revealing its boundary quotients, ideal structure, and KMS states.

Contribution

It introduces a detailed analysis of boundary quotients of the right Toeplitz algebra for the affine semigroup over natural numbers, connecting them to crossed products and group C*-algebras.

Findings

The multiplicative boundary quotient is isomorphic to a crossed product of the Toeplitz algebra of rationals.

The Crisp-Laca boundary quotient is identified as the C*-algebra of a group built from rationals.

All KMS states at inverse temperatures greater than one factor through the additive boundary quotient.

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.