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September  2021, 29(4): 2645-2656. doi: 10.3934/era.2021006

$ C^* $-algebras associated with asymptotic equivalence relations defined by hyperbolic toral automorphisms

Kengo Matsumoto

Department of Mathematics, Joetsu University of Education, Joetsu, 943-8512, Japan

Received  December 2020 Published  September 2021 Early access  January 2021

Fund Project: This work was supported by JSPS KAKENHI Grant Numbers 15K04896, 19K03537

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.

Keywords: Hyperbolic toral automorphisms, Smale space, asymptotic equivalence relation, étale groupoid, non-commutative tori.
Mathematics Subject Classification: Primary: 37D20, 37A55; Secondary: 46L35.
Citation: Kengo Matsumoto. $ C^* $-algebras associated with asymptotic equivalence relations defined by hyperbolic toral automorphisms. Electronic Research Archive, 2021, 29 (4) : 2645-2656. doi: 10.3934/era.2021006
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.  Google Scholar

[2]

C. Anantharaman-Delaroche and J. Renault, Amenable Groupoids, L'Enseignement Mathématique, Genéve, 2000.  Google Scholar

[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.  Google Scholar

[4]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math., Vol. 470. Springer-Verlag, Berlin-New York, 1975.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

K. Matsumoto, Topological conjugacy of topological Markov shifts and Ruelle algebras, J. Operator Theory, 82 (2019), 253-284.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J. Renault, A groupoid approach to $C^{\ast} $-algebras, Lecture Notes in Math., 793, Springer-Verlag, Berlin, Heidelberg and New York, 1980.  Google Scholar

[19]

J. Renault, Cartan subalgebras in $C^*$-algebras, Irish Math. Soc. Bull., 61 (2008), 29-63.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

D. Ruelle, Thermodynamic Formalism, Addison-Wesley, Reading, Mass., 1978.  Google Scholar

[23]

D. Ruelle, Non-commutative algebras for hyperbolic diffeomorphisms, Invent. Math., 93 (1988), 1-13.  doi: 10.1007/BF01393685.  Google Scholar

[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.  Google Scholar

[25]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

C. Anantharaman-Delaroche and J. Renault, Amenable Groupoids, L'Enseignement Mathématique, Genéve, 2000.  Google Scholar

[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.  Google Scholar

[4]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math., Vol. 470. Springer-Verlag, Berlin-New York, 1975.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

K. Matsumoto, Topological conjugacy of topological Markov shifts and Ruelle algebras, J. Operator Theory, 82 (2019), 253-284.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J. Renault, A groupoid approach to $C^{\ast} $-algebras, Lecture Notes in Math., 793, Springer-Verlag, Berlin, Heidelberg and New York, 1980.  Google Scholar

[19]

J. Renault, Cartan subalgebras in $C^*$-algebras, Irish Math. Soc. Bull., 61 (2008), 29-63.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

D. Ruelle, Thermodynamic Formalism, Addison-Wesley, Reading, Mass., 1978.  Google Scholar

[23]

D. Ruelle, Non-commutative algebras for hyperbolic diffeomorphisms, Invent. Math., 93 (1988), 1-13.  doi: 10.1007/BF01393685.  Google Scholar

[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.  Google Scholar

[25]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[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.  Google Scholar

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