On diagonal quasi-free automorphisms of simple Cuntz-Krieger algebras


  • Selçuk Barlak
  • Gábor Szabó




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.


Barlak, S. and Li, X., Cartan subalgebras and the UCT problem, Adv. Math. 316 (2017), 748–769. https://doi.org/10.1016/j.aim.2017.06.024

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.



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