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    Brownlowe, Nathan Ramagge, Jacqui Robertson, David and Whittaker, Michael F. 2014. Zappa–Szép products of semigroups and their -algebras. Journal of Functional Analysis, Vol. 266, Issue. 6, p. 3937.

    Brownlowe, Nathan and Raeburn, Iain 2013. Two families of Exel–Larsen crossed products. Journal of Mathematical Analysis and Applications, Vol. 398, Issue. 1, p. 68.

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    HONG, JEONG HEE LARSEN, NADIA S. and SZYMAŃSKI, WOJCIECH 2012. KMS STATES ON NICA–TOEPLITZ ALGEBRAS OF PRODUCT SYSTEMS. International Journal of Mathematics, Vol. 23, Issue. 12, p. 1250123.

    Kaliszewski, S. Landstad, Magnus B. and Quigg, John 2012. A crossed-product approach to the Cuntz–Li algebras. Proceedings of the Edinburgh Mathematical Society, Vol. 55, Issue. 02, p. 429.

    Laca, Marcelo Raeburn, Iain and Ramagge, Jacqui 2011. Phase transition on Exel crossed products associated to dilation matrices. Journal of Functional Analysis, Vol. 261, Issue. 12, p. 3633.


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

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  • Published online: 05 April 2011

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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[4]J. Crisp and M. Laca . Boundary quotients and ideals of Toeplitz C*-algebras of Artin groups. J. Funct. Anal. 242 (2007), 127156.

[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.

[12]R. Exel and A. Vershik . C*-algebras of irreversible dynamical systems. Canad. J. Math. 58 (2006), 3963.

[13]N. J. Fowler . Discrete product systems of Hilbert bimodules. Pacific J. Math. 204 (2002), 335375.

[15]M. Laca and S. Neshveyev . KMS states of quasi-free dynamics on Pimsner algebras. J. Funct. Anal. 211 (2004), 457482.

[16]M. Laca and I. Raeburn . Semigroup crossed products and the Toeplitz algebras of nonabelian groups. J. Funct. Anal. 139 (1996), 415440.

[17]M. Laca and I. Raeburn . Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers. Adv. Math. 225 (2010), 643688.

[21]J. A. Packer and M. A. Rieffel . Wavelet filter functions, the matrix completion problem, and projective modules over C(𝕋n). J. Fourier Anal. Appl. 9 (2003), 101116.

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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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