Semigroup 𝐶*-algebras of 𝑎𝑥+𝑏-semigroups

Li, Xin

doi:10.1090/tran/6469

Semigroup -algebras of -semigroups

Author: Xin Li
Journal: Trans. Amer. Math. Soc. 368 (2016), 4417-4437
MSC (2010): Primary 46L05; Secondary 11R04, 13F05
DOI: https://doi.org/10.1090/tran/6469
Published electronically: September 15, 2015
MathSciNet review: 3453375
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study semigroup -algebras of -semigroups over integral domains. The goal is to generalize several results about -algebras of -semigroups over rings of algebraic integers. We prove results concerning K-theory and structural properties like the ideal structure or pure infiniteness. Our methods allow us to treat -semigroups over a large class of integral domains containing all noetherian, integrally closed domains and coordinate rings of affine varieties over infinite fields.

[A-M] M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Company, Reading, Massachusetts, 1969.
[Bass] Hyman Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8–28. MR 0153708, https://doi.org/10.1007/BF01112819
[Bour1] N. Bourbaki, Algèbre commutative. Chapitres 1 à 4. Éléments de mathématique. Réimpression inchangée de l'édition originale de 1985, Springer-Verlag, Berlin Heidelberg, 2006.
[Bour2] N. Bourbaki, Algèbre commutative. Chapitres 5 à 7. Éléments de mathématique. Réimpression inchangée de l'édition originale de 1975, Springer-Verlag, Berlin Heidelberg, 2006.
[B-Q] Philip Quartararo Jr. and H. S. Butts, Finite unions of ideals and modules, Proc. Amer. Math. Soc. 52 (1975), 91–96. MR 0382249, https://doi.org/10.1090/S0002-9939-1975-0382249-5
[C-D-L] Joachim Cuntz, Christopher Deninger, and Marcelo Laca, 𝐶*-algebras of Toeplitz type associated with algebraic number fields, Math. Ann. 355 (2013), no. 4, 1383–1423. MR 3037019, https://doi.org/10.1007/s00208-012-0826-9
[C-E-L1] Joachim Cuntz, Siegfried Echterhoff, and Xin Li, On the K-theory of the C*-algebra generated by the left regular representation of an Ore semigroup, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 3, 645–687. MR 3323201, https://doi.org/10.4171/JEMS/513
[C-E-L2] Joachim Cuntz, Siegfried Echterhoff, and Xin Li, On the 𝐾-theory of crossed products by automorphic semigroup actions, Q. J. Math. 64 (2013), no. 3, 747–784. MR 3094498, https://doi.org/10.1093/qmath/hat021
[E-L] Siegfried Echterhoff and Marcelo Laca, The primitive ideal space of the 𝐶*-algebra of the affine semigroup of algebraic integers, Math. Proc. Cambridge Philos. Soc. 154 (2013), no. 1, 119–126. MR 3002587, https://doi.org/10.1017/S0305004112000485
[Fos] Robert M. Fossum, The divisor class group of a Krull domain, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 74. MR 0382254
[Gott] Christian Gottlieb, On finite unions of ideals and cosets, Comm. Algebra 22 (1994), no. 8, 3087–3097. MR 1272374, https://doi.org/10.1080/00927879408825014
[Hein] William Heinzer, Integral domains in which each non-zero ideal is divisorial, Mathematika 15 (1968), 164–170. MR 0236161, https://doi.org/10.1112/S0025579300002527
[H-K] Nigel Higson and Gennadi Kasparov, 𝐸-theory and 𝐾𝐾-theory for groups which act properly and isometrically on Hilbert space, Invent. Math. 144 (2001), no. 1, 23–74. MR 1821144, https://doi.org/10.1007/s002220000118
[K-R1] Eberhard Kirchberg and Mikael Rørdam, Non-simple purely infinite 𝐶*-algebras, Amer. J. Math. 122 (2000), no. 3, 637–666. MR 1759891
[K-R2] Eberhard Kirchberg and Mikael Rørdam, Infinite non-simple 𝐶*-algebras: absorbing the Cuntz algebras 𝒪_{∞}, Adv. Math. 167 (2002), no. 2, 195–264. MR 1906257, https://doi.org/10.1006/aima.2001.2041
[La-Ne] Marcelo Laca and Sergey Neshveyev, Type 𝐼𝐼𝐼₁ equilibrium states of the Toeplitz algebra of the affine semigroup over the natural numbers, J. Funct. Anal. 261 (2011), no. 1, 169–187. MR 2785897, https://doi.org/10.1016/j.jfa.2011.03.009
[L-R] Marcelo Laca and Iain Raeburn, Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers, Adv. Math. 225 (2010), no. 2, 643–688. MR 2671177, https://doi.org/10.1016/j.aim.2010.03.007
[Li1] Xin Li, Ring 𝐶*-algebras, Math. Ann. 348 (2010), no. 4, 859–898. MR 2721644, https://doi.org/10.1007/s00208-010-0502-x
[Li2] Xin Li, Semigroup 𝐶*-algebras and amenability of semigroups, J. Funct. Anal. 262 (2012), no. 10, 4302–4340. MR 2900468, https://doi.org/10.1016/j.jfa.2012.02.020
[Li3] Xin Li, Nuclearity of semigroup 𝐶*-algebras and the connection to amenability, Adv. Math. 244 (2013), 626–662. MR 3077884, https://doi.org/10.1016/j.aim.2013.05.016
[Li4] Xin Li, On K-theoretic invariants of semigroup 𝐶*-algebras attached to number fields, Adv. Math. 264 (2014), 371–395. MR 3250289, https://doi.org/10.1016/j.aim.2014.07.014
[Li-No] X. Li and M. D. Norling, Independent resolutions for totally disconnected dynamical systems II: -algebraic case, arXiv:1404.6169, accepted for publication in J. Optim. Theory.
[Nor] Magnus Dahler Norling, Inverse semigroup 𝐶*-algebras associated with left cancellative semigroups, Proc. Edinb. Math. Soc. (2) 57 (2014), no. 2, 533–564. MR 3200323, https://doi.org/10.1017/S0013091513000540
[P-R] Cornel Pasnicu and Mikael Rørdam, Purely infinite 𝐶*-algebras of real rank zero, J. Reine Angew. Math. 613 (2007), 51–73. MR 2377129, https://doi.org/10.1515/CRELLE.2007.091
[Rør] M. Rørdam, Classification of nuclear, simple 𝐶*-algebras, Classification of nuclear 𝐶*-algebras. Entropy in operator algebras, Encyclopaedia Math. Sci., vol. 126, Springer, Berlin, 2002, pp. 1–145. MR 1878882, https://doi.org/10.1007/978-3-662-04825-2_1
[Ste] Peter Stevenhagen, The arithmetic of number rings, Algorithmic number theory: lattices, number fields, curves and cryptography, Math. Sci. Res. Inst. Publ., vol. 44, Cambridge Univ. Press, Cambridge, 2008, pp. 209–266. MR 2467548