Partial crossed product description of the algebras associated with integral domains
Authors:
Giuliano Boava and Ruy Exel
Journal:
Proc. Amer. Math. Soc. 141 (2013), 24392451
MSC (2010):
Primary 46L05, 46L55
Published electronically:
April 3, 2013
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Abstract: Recently, Cuntz and Li introduced the algebra associated to an integral domain with finite quotients. In this paper, we show that is a partial group algebra of the group with suitable relations, where is the field of fractions of . We identify the spectrum of these relations and we show that it is homeomorphic to the profinite completion of . By using partial crossed product theory, we reconstruct some results proved by Cuntz and Li. Among them, we prove that is simple by showing that the action is topologically free and minimal.
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N. Brownlowe, A. an Huef, M. Laca and I. Raeburn, Boundary quotients of the Toeplitz algebra of the affine semigroup over the natural numbers, Ergodic Theory Dynam. Systems 32 (2012), no. 1, 3562. MR 2873157 (2012j:46101)
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P. B. Cohen, A dynamical system with Dedekind zeta partition function and spontaneous symmetry breaking, Journées Arithmétiques de Limoges, 1997. MR 1730430 (2001f:46104)
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J. Cuntz, algebras associated with the semigroup over , Theory and Noncommutative Geometry (Valladolid, 2006), European Math. Soc., 2008, 201215. MR 2513338 (2010i:46086)
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J. Cuntz and X. Li, The regular algebra of an integral domain, Clay Math. Proc., 11, Amer. Math. Soc., Providence, RI, 2010. MR 2732050 (2012c:46173)
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J. Cuntz and X. Li, algebras associated with integral domains and crossed products by actions on adele spaces, J. Noncomm. Geom. 5 (2011), no. 1, 137. MR 2746649 (2011k:46093)
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J. Cuntz and X. Li, Ktheory for ring algebras attached to function fields, arXiv:0911.5023v1, 2009.
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M. Dokuchaev and R. Exel, Associativity of crossed products by partial actions, enveloping actions and partial representations, Trans. Amer. Math. Soc. 357 (2005), 19311952. MR 2115083 (2005i:16066)
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R. Exel, Circle actions on algebras, partial automorphisms and a generalized PimsnerVoiculescu exact sequence, J. Funct. Analysis 122 (1994), 361401. MR 1276163 (95g:46122)
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R. Exel, Amenability for Fell bundles, J. reine angew. Math. 492 (1997), 4173. MR 1488064 (99a:46131)
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R. Exel, Partial actions of groups and actions of inverse semigroups, Proc. Amer. Math. Soc. 126 (1998), 34813494. MR 1469405 (99b:46102)
 13.
R. Exel, M. Laca and J. Quigg, Partial dynamical systems and algebras generated by partial isometries, J. Operator Theory 47 (2002), 169186. MR 1905819 (2003f:46108)
 14.
B. Julia, Statistical theory of numbers, Number Theory and Physics, Les Houches Winter School, J.M. Luck, P. Moussa and M. Waldschmidt, eds., SpringerVerlag, 1990. MR 1058473 (91h:11088)
 15.
M. Laca, Semigroups of endomorphisms, Dirichlet series and phase transitions, J. Funct. Anal. 152 (1998), 330378. MR 1608003 (99f:46097)
 16.
M. Laca and M. van Frankenhuijsen, Phase transitions on Hecke algebras and classfield theory over , J. reine angew. Math. 595 (2006), 2553. MR 2244797 (2007e:11135)
 17.
M. Laca, N. S. Larsen and S. Neshveyev, On BostConnes type systems for number fields,
J. Number Theory 129 (2009), no. 2, 325338. MR 2473881 (2010f:11156)
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M. Laca and I. Raeburn, Semigroup crossed products and the Toeplitz algebras of nonabelian groups, J. Funct. Anal. 139 (1996), 415440. MR 1402771 (97h:46109)
 19.
M. Laca and I. Raeburn, A semigroup crossed product arising in number theory, J. London Math. Soc. (2) 59 (1999), 330344. MR 1688505 (2000g:46097)
 20.
M. Laca and I. Raeburn, The ideal structure of the Hecke algebra of Bost and Connes, Math. Ann. 318 (2000), 433451. MR 1800765 (2002a:46095)
 21.
M. Laca and I. Raeburn, Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers, Adv. Math. 225 (2010), 643688. MR 2671177
 22.
N. S. Larsen and X. Li, Dilations of semigroup crossed products as crossed products of dilations, arXiv:1009.5842v1, 2010, to appear in Proc. Amer. Math. Soc.
 23.
X. Li, Ring algebras, Math. Ann. 348 (2010), no. 4, 859898. MR 2721644 (2012a:46100)
 24.
A. Nica, algebras generated by isometries and WienerHopf operators, J. Operator Theory 27 (1992), 1752. MR 1241114 (94m:46094)
 25.
S. Yamashita, Cuntz's semigroup algebra over and product system algebras, J. Ramanujan Math. Soc. 24 (2009), 299322. MR 2568059 (2010i:46089)
 1.
 J. Arledge, M. Laca and I. Raeburn, Semigroup crossed products and Hecke algebras arising from number fields, Doc. Math. 2 (1997), 115138. MR 1451963 (98k:46111)
 2.
 J. B. Bost and A. Connes, Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math., New Series, 1 (1995), no. 3, 411457. MR 1366621 (96m:46112)
 3.
 N. Brownlowe, A. an Huef, M. Laca and I. Raeburn, Boundary quotients of the Toeplitz algebra of the affine semigroup over the natural numbers, Ergodic Theory Dynam. Systems 32 (2012), no. 1, 3562. MR 2873157 (2012j:46101)
 4.
 P. B. Cohen, A dynamical system with Dedekind zeta partition function and spontaneous symmetry breaking, Journées Arithmétiques de Limoges, 1997. MR 1730430 (2001f:46104)
 5.
 J. Cuntz, algebras associated with the semigroup over , Theory and Noncommutative Geometry (Valladolid, 2006), European Math. Soc., 2008, 201215. MR 2513338 (2010i:46086)
 6.
 J. Cuntz and X. Li, The regular algebra of an integral domain, Clay Math. Proc., 11, Amer. Math. Soc., Providence, RI, 2010. MR 2732050 (2012c:46173)
 7.
 J. Cuntz and X. Li, algebras associated with integral domains and crossed products by actions on adele spaces, J. Noncomm. Geom. 5 (2011), no. 1, 137. MR 2746649 (2011k:46093)
 8.
 J. Cuntz and X. Li, Ktheory for ring algebras attached to function fields, arXiv:0911.5023v1, 2009.
 9.
 M. Dokuchaev and R. Exel, Associativity of crossed products by partial actions, enveloping actions and partial representations, Trans. Amer. Math. Soc. 357 (2005), 19311952. MR 2115083 (2005i:16066)
 10.
 R. Exel, Circle actions on algebras, partial automorphisms and a generalized PimsnerVoiculescu exact sequence, J. Funct. Analysis 122 (1994), 361401. MR 1276163 (95g:46122)
 11.
 R. Exel, Amenability for Fell bundles, J. reine angew. Math. 492 (1997), 4173. MR 1488064 (99a:46131)
 12.
 R. Exel, Partial actions of groups and actions of inverse semigroups, Proc. Amer. Math. Soc. 126 (1998), 34813494. MR 1469405 (99b:46102)
 13.
 R. Exel, M. Laca and J. Quigg, Partial dynamical systems and algebras generated by partial isometries, J. Operator Theory 47 (2002), 169186. MR 1905819 (2003f:46108)
 14.
 B. Julia, Statistical theory of numbers, Number Theory and Physics, Les Houches Winter School, J.M. Luck, P. Moussa and M. Waldschmidt, eds., SpringerVerlag, 1990. MR 1058473 (91h:11088)
 15.
 M. Laca, Semigroups of endomorphisms, Dirichlet series and phase transitions, J. Funct. Anal. 152 (1998), 330378. MR 1608003 (99f:46097)
 16.
 M. Laca and M. van Frankenhuijsen, Phase transitions on Hecke algebras and classfield theory over , J. reine angew. Math. 595 (2006), 2553. MR 2244797 (2007e:11135)
 17.
 M. Laca, N. S. Larsen and S. Neshveyev, On BostConnes type systems for number fields,
J. Number Theory 129 (2009), no. 2, 325338. MR 2473881 (2010f:11156)
 18.
 M. Laca and I. Raeburn, Semigroup crossed products and the Toeplitz algebras of nonabelian groups, J. Funct. Anal. 139 (1996), 415440. MR 1402771 (97h:46109)
 19.
 M. Laca and I. Raeburn, A semigroup crossed product arising in number theory, J. London Math. Soc. (2) 59 (1999), 330344. MR 1688505 (2000g:46097)
 20.
 M. Laca and I. Raeburn, The ideal structure of the Hecke algebra of Bost and Connes, Math. Ann. 318 (2000), 433451. MR 1800765 (2002a:46095)
 21.
 M. Laca and I. Raeburn, Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers, Adv. Math. 225 (2010), 643688. MR 2671177
 22.
 N. S. Larsen and X. Li, Dilations of semigroup crossed products as crossed products of dilations, arXiv:1009.5842v1, 2010, to appear in Proc. Amer. Math. Soc.
 23.
 X. Li, Ring algebras, Math. Ann. 348 (2010), no. 4, 859898. MR 2721644 (2012a:46100)
 24.
 A. Nica, algebras generated by isometries and WienerHopf operators, J. Operator Theory 27 (1992), 1752. MR 1241114 (94m:46094)
 25.
 S. Yamashita, Cuntz's semigroup algebra over and product system algebras, J. Ramanujan Math. Soc. 24 (2009), 299322. MR 2568059 (2010i:46089)
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Additional Information
Giuliano Boava
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada, 22460320, Rio de Janeiro/RJ, Brazil
Address at time of publication:
Departamento de Matemática, Universidade Federal de Santa Catarina, 88040900, Florianópolis/SC, Brazil
Email:
gboava@gmail.com
Ruy Exel
Affiliation:
Departamento de Matemática, Universidade Federal de Santa Catarina, 88040900, Florianópolis/SC, Brazil
Email:
r@exel.com.br
DOI:
http://dx.doi.org/10.1090/S000299392013117247
PII:
S 00029939(2013)117247
Received by editor(s):
May 23, 2011
Received by editor(s) in revised form:
October 22, 2011
Published electronically:
April 3, 2013
Additional Notes:
The first author’s research was supported by CNPq, Brazil
The second author’s research was partially supported by CNPq, Brazil
Communicated by:
Marius Junge
Article copyright:
© Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
