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

Published online by Cambridge University Press:  05 April 2011

NATHAN BROWNLOWE ,
ASTRID AN HUEF ,
MARCELO LACA  and
IAIN RAEBURN
NATHAN BROWNLOWE
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia (email: nathanb@uow.edu.au)
ASTRID AN HUEF
Affiliation:
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)
MARCELO LACA
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, Canada BC V8W 3R4 (email: laca@math.uvic.ca)
IAIN RAEBURN
Affiliation:
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)
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Abstract

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

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 1 , February 2012 , pp. 35 - 62
Copyright
Copyright © Cambridge University Press 2011

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