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

  • NATHAN BROWNLOWE (a1), ASTRID AN HUEF (a2), MARCELO LACA (a3) and IAIN RAEBURN (a2)
Abstract
Abstract

We study the Toeplitz algebra 𝒯(ℕ⋊ℕ×) and three quotients of this algebra: the C*-algebra 𝒬 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 and our new quotients are all interesting new examples for Larsen’s theory of Exel crossed products by semigroups.

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[5]J. Cuntz . C*-algebras associated with the ax+b-semigroup over ℕ. K-Theory and Noncommutative Geometry (Valladolid, 2006). European Mathematical Society, Zürich, 2008, pp. 201215.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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