Ergodic Theory and Dynamical Systems

Research Article

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

NATHAN BROWNLOWEa1, ASTRID AN HUEFa2, MARCELO LACAa3 and IAIN RAEBURNa2

a1 School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia (email: nathanb@uow.edu.au)

a2 Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand (email: astrid@maths.otago.ac.nz, iraeburn@maths.otago.ac.nz)

a3 Department of Mathematics and Statistics, University of Victoria, Victoria, Canada BC V8W 3R4 (email: laca@math.uvic.ca)

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.

(Received September 11 2010)

(Revised November 16 2010)

(Online publication April 05 2011)