Boundary quotients of the Toeplitz algebra of the affine semigroup over the natural numbers
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Abstract
Abstract We study the Toeplitz algebra 𝒯(ℕ⋊ℕ × ) and three quotients of this algebra: the C * -algebra 𝒬 ℕ recently 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⋊ℕ × 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 𝒯(ℕ⋊ℕ × ) to describe the KMS states on the two quotients. We then show that 𝒯(ℕ⋊ℕ × ), 𝒬 ℕ and our new quotients are all interesting new examples for Larsen’s theory of Exel crossed products by semigroups.
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