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

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

2010-09-20

arxiv.org/abs/1009.3678v1

Mathematics

Operator Algebras

Scientific paper

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.

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