On diagonal quasi-free automorphisms of simple Cuntz-Krieger algebras
DOI:
https://doi.org/10.7146/math.scand.a-114823Abstract
We show that an outer action of a finite abelian group on a simple Cuntz-Krieger algebra is strongly approximately inner in the sense of Izumi if the action is given by diagonal quasi-free automorphisms and the associated matrix is aperiodic. This is achieved by an approximate cohomology vanishing-type argument for the canonical shift restricted to the relative commutant of the set of domain projections of the canonical generating isometries in the fixed point algebra.
References
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Conti, R., Hong, J. H., and Szymański, W., Endomorphisms of graph algebras, J. Funct. Anal. 263 (2012), no. 9, 2529–2554. https://doi.org/10.1016/j.jfa.2012.08.024
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Cuntz, J., A class of $\mathrm C^*$-algebras and topological Markov chains. II. Reducible chains and the $\mathrm Ext$-functor for $\mathrm C^*$-algebras, Invent. Math. 63 (1981), no. 1, 25–40. https://doi.org/10.1007/BF01389192
Cuntz, J., $K$-theory for certain $\mathrm C^*$-algebras, Ann. of Math. (2) 113 (1981), no. 1, 181–197. https://doi.org/10.2307/1971137
Cuntz, J. and Krieger, W., A class of $\mathrm C^*$-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268. https://doi.org/10.1007/BF01390048
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Jeong, J. A., Purely infinite simple $C^\ast $-crossed products, Proc. Amer. Math. Soc. 123 (1995), no. 10, 3075–3078. https://doi.org/10.2307/2160662
Kirchberg, E., The classification of purely infinite $\mathrm C^*$-algebras using Kasparov's theory, Fields Institute Communications, Springer, to appear.
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Kishimoto, A., Outer automorphisms and reduced crossed products of simple $\mathrm C^*$-algebras, Comm. Math. Phys. 81 (1981), no. 3, 429–435.
Kumjian, A., Pask, D., and Raeburn, I., Cuntz-Krieger algebras of directed graphs, Pacific J. Math. 184 (1998), no. 1, 161–174. https://doi.org/10.2140/pjm.1998.184.161
Laca, M., From endomorphisms to automorphisms and back: dilations and full corners, J. London Math. Soc. (2) 61 (2000), no. 3, 893–904. https://doi.org/10.1112/S0024610799008492
Nakamura, H., Aperiodic automorphisms of nuclear purely infinite simple $\mathrm C^*$-algebras, Ergodic Theory Dynam. Systems 20 (2000), no. 6, 1749–1765. https://doi.org/10.1017/S0143385700000973
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Phillips, N. C., A classification theorem for nuclear purely infinite simple $\mathrm C^*$-algebras, Doc. Math. 5 (2000), 49–114.
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Rørdam, M., Classification of inductive limits of Cuntz algebras, J. Reine Angew. Math. 440 (1993), 175–200. https://doi.org/10.1515/crll.1993.440.175
Rørdam, M., Classification of certain infinite simple $\mathrm C^*$-algebras, J. Funct. Anal. 131 (1995), no. 2, 415–458. https://doi.org/10.1006/jfan.1995.1095
Rørdam, M., Classification of Cuntz-Krieger algebras, $K$-Theory 9 (1995), no. 1, 31–58. https://doi.org/10.1007/BF00965458
Rosenberg, J. and Schochet, C., The Künneth theorem and the universal coefficient theorem for Kasparov's generalized $K$-functor, Duke Math. J. 55 (1987), no. 2, 431–474. https://doi.org/10.1215/S0012-7094-87-05524-4
Szabó, G., Strongly self-absorbing $\mathrm C^*$-dynamical systems, Trans. Amer. Math. Soc. 370 (2018), no. 1, 99–130. https://doi.org/10.1090/tran/6931
Takai, H., On a duality for crossed products of $\mathrm C^*$-algebras, J. Functional Analysis 19 (1975), 25–39. https://doi.org/10.1016/0022-1236(75)90004-x
Toms, A. S. and Winter, W., Strongly self-absorbing $\mathrm C^*$-algebras, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3999–4029. https://doi.org/10.1090/S0002-9947-07-04173-6
Zacharias, J., Quasi-free automorphisms of Cuntz-Krieger-Pimsner algebras, in “$\mathrm C^*$-algebras (Münster, 1999)”, Springer, Berlin, 2000, pp. 262–272.
Barlak, S. and Szabó, G., Rokhlin actions of finite groups on UHF-absorbing $\mathrm C^*$-algebras, Trans. Amer. Math. Soc. 369 (2017), no. 2, 833–859. https://doi.org/10.1090/tran6697
Bratteli, O., Størmer, E., Kishimoto, A., and Rørdam, M., The crossed product of a UHF algebra by a shift, Ergodic Theory Dynam. Systems 13 (1993), no. 4, 615–626. https://doi.org/10.1017/S0143385700007574
Brown, J. and Hirshberg, I., The Rokhlin property for endomorphisms and strongly self-absorbing $\mathrm C^*$-algebras, Illinois J. Math. 58 (2014), no. 3, 619–627.
Conti, R., Hong, J. H., and Szymański, W., Endomorphisms of graph algebras, J. Funct. Anal. 263 (2012), no. 9, 2529–2554. https://doi.org/10.1016/j.jfa.2012.08.024
Cuntz, J., Automorphisms of certain simple $\mathrm C^*$-algebras, in “Quantum fields—algebras, processes (Proc. Sympos., Univ. Bielefeld, Bielefeld, 1978)”, Springer, Vienna, 1980, pp. 187–196.
Cuntz, J., A class of $\mathrm C^*$-algebras and topological Markov chains. II. Reducible chains and the $\mathrm Ext$-functor for $\mathrm C^*$-algebras, Invent. Math. 63 (1981), no. 1, 25–40. https://doi.org/10.1007/BF01389192
Cuntz, J., $K$-theory for certain $\mathrm C^*$-algebras, Ann. of Math. (2) 113 (1981), no. 1, 181–197. https://doi.org/10.2307/1971137
Cuntz, J. and Krieger, W., A class of $\mathrm C^*$-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268. https://doi.org/10.1007/BF01390048
Dadarlat, M. and Eilers, S., Asymptotic unitary equivalence in $KK$-theory, $K$-Theory 23 (2001), no. 4, 305–322. https://doi.org/10.1023/A:1011930304577
Eilers, S., Restorf, G., Ruiz, E., and Sørensen, A. P. W., The complete classification of unital graph $\mathrm C^*$-algebras: geometric and strong, preprint arXiv:1611.0712v1 [math.OA], 2016.
Evans, D. E., On $O_n$, Publ. Res. Inst. Math. Sci. 16 (1980), no. 3, 915–927. https://doi.org/10.2977/prims/1195186936
Izumi, M., Finite group actions on $\mathrm C^*$-algebras with the Rohlin property. I, Duke Math. J. 122 (2004), no. 2, 233–280. https://doi.org/10.1215/S0012-7094-04-12221-3
Izumi, M., Finite group actions on $\mathrm C^*$-algebras with the Rohlin property. II, Adv. Math. 184 (2004), no. 1, 119–160. https://doi.org/10.1016/S0001-8708(03)00140-3
Jeong, J. A., Purely infinite simple $C^\ast $-crossed products, Proc. Amer. Math. Soc. 123 (1995), no. 10, 3075–3078. https://doi.org/10.2307/2160662
Kirchberg, E., The classification of purely infinite $\mathrm C^*$-algebras using Kasparov's theory, Fields Institute Communications, Springer, to appear.
Kirchberg, E. and Phillips, N. C., Embedding of exact $\mathrm C^*$-algebras in the Cuntz algebra $\mathcal O_2$, J. Reine Angew. Math. 525 (2000), 17–53. https://doi.org/10.1515/crll.2000.065
Kishimoto, A., Outer automorphisms and reduced crossed products of simple $\mathrm C^*$-algebras, Comm. Math. Phys. 81 (1981), no. 3, 429–435.
Kumjian, A., Pask, D., and Raeburn, I., Cuntz-Krieger algebras of directed graphs, Pacific J. Math. 184 (1998), no. 1, 161–174. https://doi.org/10.2140/pjm.1998.184.161
Laca, M., From endomorphisms to automorphisms and back: dilations and full corners, J. London Math. Soc. (2) 61 (2000), no. 3, 893–904. https://doi.org/10.1112/S0024610799008492
Nakamura, H., Aperiodic automorphisms of nuclear purely infinite simple $\mathrm C^*$-algebras, Ergodic Theory Dynam. Systems 20 (2000), no. 6, 1749–1765. https://doi.org/10.1017/S0143385700000973
Pedersen, G. K., $\mathrm C^*$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc., London-New York, 1979.
Phillips, N. C., A classification theorem for nuclear purely infinite simple $\mathrm C^*$-algebras, Doc. Math. 5 (2000), 49–114.
Renault, J., Cartan subalgebras in $\mathrm C^*$-algebras, Irish Math. Soc. Bull. (2008), no. 61, 29–63.
Rørdam, M., Classification of inductive limits of Cuntz algebras, J. Reine Angew. Math. 440 (1993), 175–200. https://doi.org/10.1515/crll.1993.440.175
Rørdam, M., Classification of certain infinite simple $\mathrm C^*$-algebras, J. Funct. Anal. 131 (1995), no. 2, 415–458. https://doi.org/10.1006/jfan.1995.1095
Rørdam, M., Classification of Cuntz-Krieger algebras, $K$-Theory 9 (1995), no. 1, 31–58. https://doi.org/10.1007/BF00965458
Rosenberg, J. and Schochet, C., The Künneth theorem and the universal coefficient theorem for Kasparov's generalized $K$-functor, Duke Math. J. 55 (1987), no. 2, 431–474. https://doi.org/10.1215/S0012-7094-87-05524-4
Szabó, G., Strongly self-absorbing $\mathrm C^*$-dynamical systems, Trans. Amer. Math. Soc. 370 (2018), no. 1, 99–130. https://doi.org/10.1090/tran/6931
Takai, H., On a duality for crossed products of $\mathrm C^*$-algebras, J. Functional Analysis 19 (1975), 25–39. https://doi.org/10.1016/0022-1236(75)90004-x
Toms, A. S. and Winter, W., Strongly self-absorbing $\mathrm C^*$-algebras, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3999–4029. https://doi.org/10.1090/S0002-9947-07-04173-6
Zacharias, J., Quasi-free automorphisms of Cuntz-Krieger-Pimsner algebras, in “$\mathrm C^*$-algebras (Münster, 1999)”, Springer, Berlin, 2000, pp. 262–272.
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Published
2019-10-19
How to Cite
Barlak, S., & Szabó, G. (2019). On diagonal quasi-free automorphisms of simple Cuntz-Krieger algebras. MATHEMATICA SCANDINAVICA, 125(2), 210–226. https://doi.org/10.7146/math.scand.a-114823
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