On $C^*$-algebras associated to actions of discrete subgroups of $\operatorname{SL}(2,\mathbb{R})$ on the punctured plane
DOI:
https://doi.org/10.7146/math.scand.a-120288Abstract
Dynamical conditions that guarantee stability for discrete transformation group $C^*$-algebras are determined. The results are applied to the case of some discrete subgroups of $\operatorname{SL} (2,\mathbb{R} )$ acting on the punctured plane by means of matrix multiplication of vectors. In the case of cocompact subgroups, further properties of such crossed products are deduced from properties of the $C^*$-algebra associated to the horocycle flow on the corresponding compact homogeneous space of $\operatorname{SL} (2,\mathbb{R} )$.
References
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Anantharaman-Delaroche, C., Amenability and exactness for dynamical systems and their $C^\ast $-algebras, Trans. Amer. Math. Soc. 354 (2002), no. 10, 4153–4178. https://doi.org/10.1090/S0002-9947-02-02978-1
Antoine, R., Perera, F., and Thiel, H., Tensor products and regularity properties of Cuntz semigroups, Mem. Amer. Math. Soc. 251 (2018), no. 1199, 191 pp. https://doi.org/10.1090/memo/1199
Archbold, R. J. and Spielberg, J. S., Topologically free actions and ideals in discrete $C^*$-dynamical systems, Proc. Edinburgh Math. Soc. (2) 37 (1994), no. 1, 119–124. https://doi.org/10.1017/S0013091500018733
Bassi, J., Remarks on some simple $C^*$-algebras admitting a unique lower semicontinuous $2$-quasitrace, eprint arXiv:1806.08288 [math.OA], 2018.
Bekka, M. B. and Mayer, M., Ergodic theory and topological dynamics of group actions on homogeneous spaces, London Mathematical Society Lecture Note Series, vol. 269, Cambridge University Press, Cambridge, 2000. https://doi.org/10.1017/CBO9780511758898
Blackadar, B., Robert, L., Tikuisis, A. P., Toms, A. S., and Winter, W., An algebraic approach to the radius of comparison, Trans. Amer. Math. Soc. 364 (2012), no. 7, 3657–3674. https://doi.org/10.1090/S0002-9947-2012-05538-3
Blackadar, B. E. and Cuntz, J., The structure of stable algebraically simple $C^\ast $-algebras, Amer. J. Math. 104 (1982), no. 4, 813–822. https://doi.org/10.2307/2374206
Connes, A., An analogue of the Thom isomorphism for crossed products of a $C^\ast $-algebra by an action of ℝ, Adv. in Math. 39 (1981), no. 1, 31–55. https://doi.org/10.1016/0001-8708(81)90056-6
Connes, A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
Coward, K. T., Elliott, G. A., and Ivanescu, C., The Cuntz semigroup as an invariant for $C^*$-algebras, J. Reine Angew. Math. 623 (2008), 161–193. https://doi.org/10.1515/CRELLE.2008.075
Dixmier, J., $C^*$-algebras, North-Holland Mathematical Library, vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
Einsiedler, M. and Ward, T., Ergodic theory with a view towards number theory, Graduate Texts in Mathematics, vol. 259, Springer-Verlag London, Ltd., London, 2011. https://doi.org/10.1007/978-0-85729-021-2
Furstenberg, H., The unique ergodicity of the horocycle flow, in “Recent advances in topological dynamics (Proc. Conf., Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund)”, Lecture Notes in Math., vol. 318, 1973, pp. 95–115.
Giordano, T., Putnam, I. F., and Skau, C. F., Topological orbit equivalence and $C^*$-crossed products, J. Reine Angew. Math. 469 (1995), 51–111.
Glasner, S., Proximal flows, Lecture Notes in Mathematics, Vol. 517, Springer-Verlag, Berlin-New York, 1976.
Gorodnik, A. and Weiss, B., Distribution of lattice orbits on homogeneous varieties, Geom. Funct. Anal. 17 (2007), no. 1, 58–115. https://doi.org/10.1007/s00039-006-0583-6
Green, P., $C^*$-algebras of transformation groups with smooth orbit space, Pacific J. Math. 72 (1977), no. 1, 71–97.
Green, P., The local structure of twisted covariance algebras, Acta Math. 140 (1978), no. 3-4, 191–250. https://doi.org/10.1007/BF02392308
Guilloux, A., A brief remark on orbits of $\operatorname SL(2,\mathbb Z)$ in the Euclidean plane, Ergodic Theory Dynam. Systems 30 (2010), no. 4, 1101–1109. https://doi.org/10.1017/S0143385709000315
Hirshberg, I., Szabó, G., Winter, W., and Wu, J., Rokhlin dimension for flows, Comm. Math. Phys. 353 (2017), no. 1, 253–316. https://doi.org/10.1007/s00220-016-2762-0
Hirshberg, I., Winter, W., and Zacharias, J., Rokhlin dimension and $C^*$-dynamics, Comm. Math. Phys. 335 (2015), no. 2, 637–670. https://doi.org/10.1007/s00220-014-2264-x
Katok, S., Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992.
Kirchberg, E., On the existence of traces on exact stably projectionless simple $C^\ast $-algebras, in “Operator algebras and their applications (Waterloo, ON, 1994/1995)”, Fields Inst. Commun., vol. 13, Amer. Math. Soc., Providence, RI, 1997, pp. 171–172.
Kra, I., On lifting Kleinian groups to $\operatorname SL(2,\mathbb C)$, in “Differential geometry and complex analysis”, Springer, Berlin, 1985, pp. 181–193.
Laca, M. and Spielberg, J., Purely infinite $C^*$-algebras from boundary actions of discrete groups, J. Reine Angew. Math. 480 (1996), 125–139. https://doi.org/10.1515/crll.1996.480.125
Ledrappier, F., Distribution des orbites des réseaux sur le plan réel, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 1, 61–64. https://doi.org/10.1016/S0764-4442(99)80462-5
Maucourant, F. and Weiss, B., Lattice actions on the plane revisited, Geom. Dedicata 157 (2012), 1–21. https://doi.org/10.1007/s10711-011-9596-x
Nogueira, A., Orbit distribution on $\mathbb R^2$ under the natural action of $\operatorname SL(2,\mathbb Z)$, Indag. Math. (N.S.) 13 (2002), no. 1, 103–124. https://doi.org/10.1016/S0019-3577(02)90009-1
Nogueira, A., Lattice orbit distribution on $\mathbb R^2$, Ergodic Theory Dynam. Systems 30 (2010), no. 4, 1201–1214. https://doi.org/10.1017/S0143385709000558
Ortega, E., Rørdam, M., and Thiel, H., The Cuntz semigroup and comparison of open projections, J. Funct. Anal. 260 (2011), no. 12, 3474–3493. https://doi.org/10.1016/j.jfa.2011.02.017
Paternain, G. P., Magnetic rigidity of horocycle flows, Pacific J. Math. 225 (2006), no. 2, 301–323. https://doi.org/10.2140/pjm.2006.225.301
Poon, Y. T., Stable rank of some crossed product $C^*$-algebras, Proc. Amer. Math. Soc. 105 (1989), no. 4, 868–875. https://doi.org/10.2307/2047045
Rieffel, M. A., $C^\ast $-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), no. 2, 415–429.
Rieffel, M. A., Dimension and stable rank in the $K$-theory of $C^\ast $-algebras, Proc. London Math. Soc. (3) 46 (1983), no. 2, 301–333. https://doi.org/10.1112/plms/s3-46.2.301
Robert, L., Remarks on $\mathcal Z$-stable projectionless $\mathrm C^*$-algebras, Glasg. Math. J. 58 (2016), no. 2, 273–277. https://doi.org/10.1017/S0017089515000117
Rørdam, M., On the structure of simple $C^*$-algebras tensored with a UHF-algebra, J. Funct. Anal. 100 (1991), no. 1, 1–17. https://doi.org/10.1016/0022-1236(91)90098-P
Rørdam, M., On sums of finite projections, in “Operator algebras and operator theory (Shanghai, 1997)”, Contemp. Math., vol. 228, Amer. Math. Soc., Providence, RI, 1998, pp. 327–340. https://doi.org/10.1090/conm/228/03295
Rørdam, M., The stable and the real rank of $\mathcal Z$-absorbing $C^*$-algebras, Internat. J. Math. 15 (2004), no. 10, 1065–1084. https://doi.org/10.1142/S0129167X04002661
Semenova, O. L., A property of purely hyperbolic Fuchsian groups of the second kind, J. Math. Sci. 107 (2001), no. 4, 4092–4096. https://doi.org/10.1023/A:1012448902514
Sierakowski, A., Discrete crossed product $C^*$-algebras, Ph.D. thesis, University of Copenhagen, 2009.
Szabó, G., The Rokhlin dimension of topological $\mathbb Z^m$-actions, Proc. Lond. Math. Soc. (3) 110 (2015), no. 3, 673–694. https://doi.org/10.1112/plms/pdu065
Szabó, G., Wu, J., and Zacharias, J., Rokhlin dimension for actions of residually finite groups, Ergodic Theory Dynam. Systems 39 (2019), no. 8, 2248–2304. https://doi.org/10.1017/etds.2017.113
Tikuisis, A., Nuclear dimension, $\mathcal Z$-stability, and algebraic simplicity for stably projectionless $C^\ast $-algebras, Math. Ann. 358 (2014), no. 3-4, 729–778. https://doi.org/10.1007/s00208-013-0951-0
Toms, A. S. and Winter, W., Minimal dynamics and the classification of $C^*$-algebras, Proc. Natl. Acad. Sci. USA 106 (2009), no. 40, 16942–16943. https://doi.org/10.1073/pnas.0903629106
Williams, D. P., Crossed products of $C^\ast $-algebras, Mathematical Surveys and Monographs, vol. 134, American Mathematical Society, Providence, RI, 2007. https://doi.org/10.1090/surv/134
Anantharaman-Delaroche, C., Amenability and exactness for dynamical systems and their $C^\ast $-algebras, Trans. Amer. Math. Soc. 354 (2002), no. 10, 4153–4178. https://doi.org/10.1090/S0002-9947-02-02978-1
Antoine, R., Perera, F., and Thiel, H., Tensor products and regularity properties of Cuntz semigroups, Mem. Amer. Math. Soc. 251 (2018), no. 1199, 191 pp. https://doi.org/10.1090/memo/1199
Archbold, R. J. and Spielberg, J. S., Topologically free actions and ideals in discrete $C^*$-dynamical systems, Proc. Edinburgh Math. Soc. (2) 37 (1994), no. 1, 119–124. https://doi.org/10.1017/S0013091500018733
Bassi, J., Remarks on some simple $C^*$-algebras admitting a unique lower semicontinuous $2$-quasitrace, eprint arXiv:1806.08288 [math.OA], 2018.
Bekka, M. B. and Mayer, M., Ergodic theory and topological dynamics of group actions on homogeneous spaces, London Mathematical Society Lecture Note Series, vol. 269, Cambridge University Press, Cambridge, 2000. https://doi.org/10.1017/CBO9780511758898
Blackadar, B., Robert, L., Tikuisis, A. P., Toms, A. S., and Winter, W., An algebraic approach to the radius of comparison, Trans. Amer. Math. Soc. 364 (2012), no. 7, 3657–3674. https://doi.org/10.1090/S0002-9947-2012-05538-3
Blackadar, B. E. and Cuntz, J., The structure of stable algebraically simple $C^\ast $-algebras, Amer. J. Math. 104 (1982), no. 4, 813–822. https://doi.org/10.2307/2374206
Connes, A., An analogue of the Thom isomorphism for crossed products of a $C^\ast $-algebra by an action of ℝ, Adv. in Math. 39 (1981), no. 1, 31–55. https://doi.org/10.1016/0001-8708(81)90056-6
Connes, A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
Coward, K. T., Elliott, G. A., and Ivanescu, C., The Cuntz semigroup as an invariant for $C^*$-algebras, J. Reine Angew. Math. 623 (2008), 161–193. https://doi.org/10.1515/CRELLE.2008.075
Dixmier, J., $C^*$-algebras, North-Holland Mathematical Library, vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
Einsiedler, M. and Ward, T., Ergodic theory with a view towards number theory, Graduate Texts in Mathematics, vol. 259, Springer-Verlag London, Ltd., London, 2011. https://doi.org/10.1007/978-0-85729-021-2
Furstenberg, H., The unique ergodicity of the horocycle flow, in “Recent advances in topological dynamics (Proc. Conf., Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund)”, Lecture Notes in Math., vol. 318, 1973, pp. 95–115.
Giordano, T., Putnam, I. F., and Skau, C. F., Topological orbit equivalence and $C^*$-crossed products, J. Reine Angew. Math. 469 (1995), 51–111.
Glasner, S., Proximal flows, Lecture Notes in Mathematics, Vol. 517, Springer-Verlag, Berlin-New York, 1976.
Gorodnik, A. and Weiss, B., Distribution of lattice orbits on homogeneous varieties, Geom. Funct. Anal. 17 (2007), no. 1, 58–115. https://doi.org/10.1007/s00039-006-0583-6
Green, P., $C^*$-algebras of transformation groups with smooth orbit space, Pacific J. Math. 72 (1977), no. 1, 71–97.
Green, P., The local structure of twisted covariance algebras, Acta Math. 140 (1978), no. 3-4, 191–250. https://doi.org/10.1007/BF02392308
Guilloux, A., A brief remark on orbits of $\operatorname SL(2,\mathbb Z)$ in the Euclidean plane, Ergodic Theory Dynam. Systems 30 (2010), no. 4, 1101–1109. https://doi.org/10.1017/S0143385709000315
Hirshberg, I., Szabó, G., Winter, W., and Wu, J., Rokhlin dimension for flows, Comm. Math. Phys. 353 (2017), no. 1, 253–316. https://doi.org/10.1007/s00220-016-2762-0
Hirshberg, I., Winter, W., and Zacharias, J., Rokhlin dimension and $C^*$-dynamics, Comm. Math. Phys. 335 (2015), no. 2, 637–670. https://doi.org/10.1007/s00220-014-2264-x
Katok, S., Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992.
Kirchberg, E., On the existence of traces on exact stably projectionless simple $C^\ast $-algebras, in “Operator algebras and their applications (Waterloo, ON, 1994/1995)”, Fields Inst. Commun., vol. 13, Amer. Math. Soc., Providence, RI, 1997, pp. 171–172.
Kra, I., On lifting Kleinian groups to $\operatorname SL(2,\mathbb C)$, in “Differential geometry and complex analysis”, Springer, Berlin, 1985, pp. 181–193.
Laca, M. and Spielberg, J., Purely infinite $C^*$-algebras from boundary actions of discrete groups, J. Reine Angew. Math. 480 (1996), 125–139. https://doi.org/10.1515/crll.1996.480.125
Ledrappier, F., Distribution des orbites des réseaux sur le plan réel, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 1, 61–64. https://doi.org/10.1016/S0764-4442(99)80462-5
Maucourant, F. and Weiss, B., Lattice actions on the plane revisited, Geom. Dedicata 157 (2012), 1–21. https://doi.org/10.1007/s10711-011-9596-x
Nogueira, A., Orbit distribution on $\mathbb R^2$ under the natural action of $\operatorname SL(2,\mathbb Z)$, Indag. Math. (N.S.) 13 (2002), no. 1, 103–124. https://doi.org/10.1016/S0019-3577(02)90009-1
Nogueira, A., Lattice orbit distribution on $\mathbb R^2$, Ergodic Theory Dynam. Systems 30 (2010), no. 4, 1201–1214. https://doi.org/10.1017/S0143385709000558
Ortega, E., Rørdam, M., and Thiel, H., The Cuntz semigroup and comparison of open projections, J. Funct. Anal. 260 (2011), no. 12, 3474–3493. https://doi.org/10.1016/j.jfa.2011.02.017
Paternain, G. P., Magnetic rigidity of horocycle flows, Pacific J. Math. 225 (2006), no. 2, 301–323. https://doi.org/10.2140/pjm.2006.225.301
Poon, Y. T., Stable rank of some crossed product $C^*$-algebras, Proc. Amer. Math. Soc. 105 (1989), no. 4, 868–875. https://doi.org/10.2307/2047045
Rieffel, M. A., $C^\ast $-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), no. 2, 415–429.
Rieffel, M. A., Dimension and stable rank in the $K$-theory of $C^\ast $-algebras, Proc. London Math. Soc. (3) 46 (1983), no. 2, 301–333. https://doi.org/10.1112/plms/s3-46.2.301
Robert, L., Remarks on $\mathcal Z$-stable projectionless $\mathrm C^*$-algebras, Glasg. Math. J. 58 (2016), no. 2, 273–277. https://doi.org/10.1017/S0017089515000117
Rørdam, M., On the structure of simple $C^*$-algebras tensored with a UHF-algebra, J. Funct. Anal. 100 (1991), no. 1, 1–17. https://doi.org/10.1016/0022-1236(91)90098-P
Rørdam, M., On sums of finite projections, in “Operator algebras and operator theory (Shanghai, 1997)”, Contemp. Math., vol. 228, Amer. Math. Soc., Providence, RI, 1998, pp. 327–340. https://doi.org/10.1090/conm/228/03295
Rørdam, M., The stable and the real rank of $\mathcal Z$-absorbing $C^*$-algebras, Internat. J. Math. 15 (2004), no. 10, 1065–1084. https://doi.org/10.1142/S0129167X04002661
Semenova, O. L., A property of purely hyperbolic Fuchsian groups of the second kind, J. Math. Sci. 107 (2001), no. 4, 4092–4096. https://doi.org/10.1023/A:1012448902514
Sierakowski, A., Discrete crossed product $C^*$-algebras, Ph.D. thesis, University of Copenhagen, 2009.
Szabó, G., The Rokhlin dimension of topological $\mathbb Z^m$-actions, Proc. Lond. Math. Soc. (3) 110 (2015), no. 3, 673–694. https://doi.org/10.1112/plms/pdu065
Szabó, G., Wu, J., and Zacharias, J., Rokhlin dimension for actions of residually finite groups, Ergodic Theory Dynam. Systems 39 (2019), no. 8, 2248–2304. https://doi.org/10.1017/etds.2017.113
Tikuisis, A., Nuclear dimension, $\mathcal Z$-stability, and algebraic simplicity for stably projectionless $C^\ast $-algebras, Math. Ann. 358 (2014), no. 3-4, 729–778. https://doi.org/10.1007/s00208-013-0951-0
Toms, A. S. and Winter, W., Minimal dynamics and the classification of $C^*$-algebras, Proc. Natl. Acad. Sci. USA 106 (2009), no. 40, 16942–16943. https://doi.org/10.1073/pnas.0903629106
Williams, D. P., Crossed products of $C^\ast $-algebras, Mathematical Surveys and Monographs, vol. 134, American Mathematical Society, Providence, RI, 2007. https://doi.org/10.1090/surv/134
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2020-09-03
How to Cite
Bassi, J. (2020). On $C^*$-algebras associated to actions of discrete subgroups of $\operatorname{SL}(2,\mathbb{R})$ on the punctured plane. MATHEMATICA SCANDINAVICA, 126(3), 540–558. https://doi.org/10.7146/math.scand.a-120288
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