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

TL;DR

This paper investigates the structure and KMS states of boundary quotients of the Toeplitz algebra associated with the affine semigroup over natural numbers, introducing new quotients and analyzing their properties.

Contribution

It introduces two new boundary quotients of the Toeplitz algebra, characterizes their structure as partial crossed products, and analyzes their KMS states.

Findings

The additive boundary quotient satisfies a uniqueness theorem.

KMS states on the quotients are explicitly described.

The algebras are examples in Exel crossed product theory.

Abstract

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.