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$ C^* $-algebras associated with asymptotic equivalence relations defined by hyperbolic toral automorphisms
Department of Mathematics, Joetsu University of Education, Joetsu, 943-8512, Japan |
We study the $ C^* $-algebras of the étale groupoids defined by the asymptotic equivalence relations for hyperbolic automorphisms on the two-dimensional torus. The algebras are proved to be isomorphic to four-dimensional non-commutative tori by an explicit numerical computation. The ranges of the unique tracial states of its $ K_0 $-groups of the $ C^* $-algebras are described in terms of the hyperbolic matrices of the automorphisms on the torus.
References:
[1] |
R. L. Adler and R. Palais,
Homeomorphic conjugacy of automorphisms of the torus, Proc. Amer. Math. Soc., 16 (1965), 1222-1225.
doi: 10.1090/S0002-9939-1965-0193181-8. |
[2] |
C. Anantharaman-Delaroche and J. Renault, Amenable Groupoids, L'Enseignement Mathématique, Genéve, 2000. |
[3] |
F. P. Boca,
The structure of higher-dimensional non-commutative tori and metric Diophantine approximation, J. Reine Angew. Math., 492 (1997), 179-219.
doi: 10.1515/crll.1997.492.179. |
[4] |
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math., Vol. 470. Springer-Verlag, Berlin-New York, 1975. |
[5] |
J. Cuntz and W. Krieger,
A class of $C^{\ast} $-algebras and topological Markov chains, Invent. Math., 56 (1980), 251-268.
doi: 10.1007/BF01390048. |
[6] |
G. A. Elliott, On the $K$-theory of the $C^{\ast} $–algebra generated by a projective representation of a torsion-free discrete abelian group, in Operator Algebras and Group Representations, Pitman, Boston, MA, 17 (1984), 157–184. |
[7] |
C. G. Holton,
The Rohlin property for shifts of finite type, J. Funct. Anal., 229 (2005), 277-299.
doi: 10.1016/j.jfa.2005.05.002. |
[8] |
J. Kaminker and I. Putnam,
$K$-theoretic duality of shifts of finite type, Comm. Math. Phys., 187 (1997), 509-522.
doi: 10.1007/s002200050147. |
[9] |
J. Kaminker, I. Putnam and J. Spielberg, Operator algebras and hyperbolic dynamics, Operator Algebras and Quantum Field Theory, 525-532, Int. Press, Cambridge, MA, 1997. |
[10] |
D. B. Killough and I. F. Putnam,
Ring and module structures on dimension groups associated with a shift of finite type, Ergodic Theory Dynam. Systems, 32 (2012), 1370-1399.
doi: 10.1017/S0143385712000272. |
[11] |
K. Matsumoto,
Asymptotic continuous orbit equivalence of Smale spaces and Ruelle algebras, Canad. J. Math., 71 (2019), 1243-1296.
doi: 10.4153/CJM-2018-012-x. |
[12] |
K. Matsumoto,
Topological conjugacy of topological Markov shifts and Ruelle algebras, J. Operator Theory, 82 (2019), 253-284.
|
[13] |
N. C. Phillips, Every simple higher dimensional non-commutative torus is an AT algebra, preprint, arXiv: math.OA/0609783. Google Scholar |
[14] |
I. F. Putnam,
$C^*$-algebras from Smale spaces, Canad. J. Math., 48 (1996), 175-195.
doi: 10.4153/CJM-1996-008-2. |
[15] |
I. F. Putnam, Hyperbolic Systems and Generalized Cuntz–Krieger Algebras, Lecture Notes, Summer School in Operator Algebras, Odense August 1996. Google Scholar |
[16] |
I. F. Putnam, A homology theory for Smale spaces, Mem. Amer. Math. Soc. 232 (2014), No. 1094.
doi: 10.1090/memo/1094. |
[17] |
I. F. Putnam and J. Spielberg,
The structure of $C^*$-algebras associated with hyperbolic dynamical systems, J. Funct. Anal., 163 (1999), 279-299.
doi: 10.1006/jfan.1998.3379. |
[18] |
J. Renault, A groupoid approach to $C^{\ast} $-algebras, Lecture Notes in Math., 793, Springer-Verlag, Berlin, Heidelberg and New York, 1980. |
[19] |
J. Renault,
Cartan subalgebras in $C^*$-algebras, Irish Math. Soc. Bull., 61 (2008), 29-63.
|
[20] |
M. A. Rieffel,
$C^{\ast} $-algebras associated with irrational rotations, Pacific J. Math., 93 (1981), 415-429.
doi: 10.2140/pjm.1981.93.415. |
[21] |
M. A. Rieffel,
Projective modules over higher-dimensional non-commutative tori, Canad. J. Math., 40 (1988), 257-338.
doi: 10.4153/CJM-1988-012-9. |
[22] |
D. Ruelle, Thermodynamic Formalism, Addison-Wesley, Reading, Mass., 1978. |
[23] |
D. Ruelle,
Non-commutative algebras for hyperbolic diffeomorphisms, Invent. Math., 93 (1988), 1-13.
doi: 10.1007/BF01393685. |
[24] |
J. Slawny,
On factor representations and the $C^{\ast} $-algebra of canonical commutation relations, Comm. Math. Phys., 24 (1972), 151-170.
doi: 10.1007/BF01878451. |
[25] |
S. Smale,
Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.
doi: 10.1090/S0002-9904-1967-11798-1. |
[26] |
K. Thomsen, $C^*$-algebras of homoclinic and heteroclinic structure in expansive dynamics, Mem. Amer. Math. Soc., 206 (2010), No. 970.
doi: 10.1090/S0065-9266-10-00581-8. |
show all references
References:
[1] |
R. L. Adler and R. Palais,
Homeomorphic conjugacy of automorphisms of the torus, Proc. Amer. Math. Soc., 16 (1965), 1222-1225.
doi: 10.1090/S0002-9939-1965-0193181-8. |
[2] |
C. Anantharaman-Delaroche and J. Renault, Amenable Groupoids, L'Enseignement Mathématique, Genéve, 2000. |
[3] |
F. P. Boca,
The structure of higher-dimensional non-commutative tori and metric Diophantine approximation, J. Reine Angew. Math., 492 (1997), 179-219.
doi: 10.1515/crll.1997.492.179. |
[4] |
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math., Vol. 470. Springer-Verlag, Berlin-New York, 1975. |
[5] |
J. Cuntz and W. Krieger,
A class of $C^{\ast} $-algebras and topological Markov chains, Invent. Math., 56 (1980), 251-268.
doi: 10.1007/BF01390048. |
[6] |
G. A. Elliott, On the $K$-theory of the $C^{\ast} $–algebra generated by a projective representation of a torsion-free discrete abelian group, in Operator Algebras and Group Representations, Pitman, Boston, MA, 17 (1984), 157–184. |
[7] |
C. G. Holton,
The Rohlin property for shifts of finite type, J. Funct. Anal., 229 (2005), 277-299.
doi: 10.1016/j.jfa.2005.05.002. |
[8] |
J. Kaminker and I. Putnam,
$K$-theoretic duality of shifts of finite type, Comm. Math. Phys., 187 (1997), 509-522.
doi: 10.1007/s002200050147. |
[9] |
J. Kaminker, I. Putnam and J. Spielberg, Operator algebras and hyperbolic dynamics, Operator Algebras and Quantum Field Theory, 525-532, Int. Press, Cambridge, MA, 1997. |
[10] |
D. B. Killough and I. F. Putnam,
Ring and module structures on dimension groups associated with a shift of finite type, Ergodic Theory Dynam. Systems, 32 (2012), 1370-1399.
doi: 10.1017/S0143385712000272. |
[11] |
K. Matsumoto,
Asymptotic continuous orbit equivalence of Smale spaces and Ruelle algebras, Canad. J. Math., 71 (2019), 1243-1296.
doi: 10.4153/CJM-2018-012-x. |
[12] |
K. Matsumoto,
Topological conjugacy of topological Markov shifts and Ruelle algebras, J. Operator Theory, 82 (2019), 253-284.
|
[13] |
N. C. Phillips, Every simple higher dimensional non-commutative torus is an AT algebra, preprint, arXiv: math.OA/0609783. Google Scholar |
[14] |
I. F. Putnam,
$C^*$-algebras from Smale spaces, Canad. J. Math., 48 (1996), 175-195.
doi: 10.4153/CJM-1996-008-2. |
[15] |
I. F. Putnam, Hyperbolic Systems and Generalized Cuntz–Krieger Algebras, Lecture Notes, Summer School in Operator Algebras, Odense August 1996. Google Scholar |
[16] |
I. F. Putnam, A homology theory for Smale spaces, Mem. Amer. Math. Soc. 232 (2014), No. 1094.
doi: 10.1090/memo/1094. |
[17] |
I. F. Putnam and J. Spielberg,
The structure of $C^*$-algebras associated with hyperbolic dynamical systems, J. Funct. Anal., 163 (1999), 279-299.
doi: 10.1006/jfan.1998.3379. |
[18] |
J. Renault, A groupoid approach to $C^{\ast} $-algebras, Lecture Notes in Math., 793, Springer-Verlag, Berlin, Heidelberg and New York, 1980. |
[19] |
J. Renault,
Cartan subalgebras in $C^*$-algebras, Irish Math. Soc. Bull., 61 (2008), 29-63.
|
[20] |
M. A. Rieffel,
$C^{\ast} $-algebras associated with irrational rotations, Pacific J. Math., 93 (1981), 415-429.
doi: 10.2140/pjm.1981.93.415. |
[21] |
M. A. Rieffel,
Projective modules over higher-dimensional non-commutative tori, Canad. J. Math., 40 (1988), 257-338.
doi: 10.4153/CJM-1988-012-9. |
[22] |
D. Ruelle, Thermodynamic Formalism, Addison-Wesley, Reading, Mass., 1978. |
[23] |
D. Ruelle,
Non-commutative algebras for hyperbolic diffeomorphisms, Invent. Math., 93 (1988), 1-13.
doi: 10.1007/BF01393685. |
[24] |
J. Slawny,
On factor representations and the $C^{\ast} $-algebra of canonical commutation relations, Comm. Math. Phys., 24 (1972), 151-170.
doi: 10.1007/BF01878451. |
[25] |
S. Smale,
Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.
doi: 10.1090/S0002-9904-1967-11798-1. |
[26] |
K. Thomsen, $C^*$-algebras of homoclinic and heteroclinic structure in expansive dynamics, Mem. Amer. Math. Soc., 206 (2010), No. 970.
doi: 10.1090/S0065-9266-10-00581-8. |
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