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


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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[1]Brownlowe N.. Realising the C *-algebra of a higher-rank graph as an Exel crossed product. J. Operator Theory to appear; arXiv:0912.4635.
[2]Brownlowe N. and Raeburn I.. Exel’s crossed product and relative Cuntz–Pimsner algebras. Math. Proc. Cambridge Philos. Soc. 141 (2006), 497508.
[3]Connes A. and Marcolli M.. Noncommutative Geometry, Quantum Fields, and Motives (Colloquium Publications, 55). American Mathematical Society, Providence, RI, 2008.
[4]Crisp J. and Laca M.. Boundary quotients and ideals of Toeplitz C *-algebras of Artin groups. J. Funct. Anal. 242 (2007), 127156.
[5]Cuntz J.. C *-algebras associated with the ax+b-semigroup over ℕ. K-Theory and Noncommutative Geometry (Valladolid, 2006). European Mathematical Society, Zürich, 2008, pp. 201215.
[6]Echterhoff S., Kaliszewski S., Quigg J. and Raeburn I.. A categorical approach to imprimitivity theorems for C*-dynamical systems. Mem. Amer. Math. Soc. 180(850) (2006), viii+169.
[7]Exel R.. Amenability for Fell bundles. J. Reine Angew. Math. 492 (1997), 4173.
[8]Exel R.. A new look at the crossed-product of a C *-algebra by an endomorphism. Ergod. Th. & Dynam. Sys. 23 (2003), 118.
[9]Exel R.. A new look at the crossed-product of a C *-algebra by a semigroup of endomorphisms. Ergod. Th. & Dynam. Sys. 28 (2008), 749789.
[10]Exel R., an Huef A. and Raeburn I.. Purely infinite simple C *-algebras associated to integer dilation matrices, Indiana Univ. Math. J. to appear.
[11]Exel R., Laca M. and Quigg J.. Partial dynamical systems and C *-algebras generated by partial isometries. J. Operator Theory 47 (2002), 169186.
[12]Exel R. and Vershik A.. C *-algebras of irreversible dynamical systems. Canad. J. Math. 58 (2006), 3963.
[13]Fowler N. J.. Discrete product systems of Hilbert bimodules. Pacific J. Math. 204 (2002), 335375.
[14]Laca M.. Purely infinite simple Toeplitz algebras. J. Operator Theory 41 (1999), 421435.
[15]Laca M. and Neshveyev S.. KMS states of quasi-free dynamics on Pimsner algebras. J. Funct. Anal. 211 (2004), 457482.
[16]Laca M. and Raeburn I.. Semigroup crossed products and the Toeplitz algebras of nonabelian groups. J. Funct. Anal. 139 (1996), 415440.
[17]Laca M. and Raeburn I.. Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers. Adv. Math. 225 (2010), 643688.
[18]Larsen N. S.. Crossed products by abelian semigroups via transfer operators. Ergod. Th. & Dynam. Sys. 30 (2010), 11471164.
[19]Larsen N. S. and Raeburn I.. Projective multi-resolution analyses arising from direct limits of Hilbert modules. Math. Scand. 100 (2007), 317360.
[20]Nica A.. C *-algebras generated by isometries and Wiener–Hopf operators. J. Operator Theory 27 (1992), 1752.
[21]Packer J. A. and Rieffel M. A.. Wavelet filter functions, the matrix completion problem, and projective modules over C(𝕋n). J. Fourier Anal. Appl. 9 (2003), 101116.
[22]Sims A. and Yeend T.. C *-algebras associated to product systems of Hilbert bimodules. J. Operator Theory 64 (2010), 349376.
[23]Yamashita S.. Cuntz’s ax+b-semigroup C *-algebra over ℕ and product system C *-algebras. J. Ramanujan Math. Soc. 24 (2009), 299322.
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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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