Localized Automorphisms and Endomorphisms


  • Abdelgabar Adam Hassan Department of Mathematics, College of Science, Jouf University, Kingdom of Saudi Arabia; Department of Mathematics, University of Nyala, Nyala, Sudan https://orcid.org/0000-0001-6233-2332




Algebra, Endomorphism, Automorphism, Uniformly hyperfinite, Cuntz algebra


We give a practical criterion of invertibility of endomorphisms of \(O_n\) corresponding to unitaries in the normalizer of the diagonal inside the uniformly hyperfinite subalgebra. We also analyze the action of such localized automorphisms on the spectrum of the diagonal thus obtaining criteria of outerness. Unital endomorphisms of the Cuntz algebra \(O_n\) which preserve the canonical uniformly hyperfinite-subalgebra \(F_n \subseteq O_n\) are investigated. We give examples of such endomorphisms \(\lambda =\lambda_u\) for which the associated unitary element \(u\) in \(O_n\).


Download data is not yet available.


R. Conti, Automorphisms of the UHF algebra that do not extend to the Cuntz algebra, Journal of the Australian Mathematical Society 89(3) (2010), 309 – 315, DOI: 10.1017/S1446788711001017.

R. Conti and W. Szymanski, Automorphisms of the Cuntz algebras, in: The Proceedings of the EU-NCG Meeting in Bucharest, Bucharest (2011).

R. Conti and W. Szymanski, Labeled trees and localized automorphisms of the Cuntz algebras, Transactions of the American Mathematical Society 363(11) (2011), 5847 – 5870, DOI: 10.1090/S0002-9947-2011-05234-7.

R. Conti, J.H. Hong and W. Szymanski, Endomorphisms of graph algebras, Journal of Functional Analysis 263(9) (2012), 2529 – 2554, DOI: 10.1016/j.jfa.2012.08.024.

R. Conti, J.H. Hong and W. Szymanski, The restricted Weyl group of the Cuntz algebra and shift endomorphisms, Journal für die reine und angewandte Mathematik 667 (2012), 177 – 191, DOI: 10.1515/CRELLE.2011.125.

R. Conti, M. Rørdam and W. Szymanski, Endomorphisms of On which preserve the canonical UHF subalgebra, Journal of Functional Analysis 259 (2010), 602 – 617, DOI: 10.1016/j.jfa.2010.03.027.

J. Cuntz, A class of C∗-algebras and topological Markov chains II: Reducible chains and the Ext function for C∗-algebras, Inventiones Mathematicae 63 (1981), 25 – 40, DOI: 10.1007/BF01389192.

J. Cuntz, Automorphisms of certain simple C∗-algebras, in: Quantum Fields – Algebras, Processes, L. Streit (editor), Springer, Vienna (1980), DOI: 10.1007/978-3-7091-8598-8_13.

J. Cuntz, Simple C∗-algebras generated by isometries, Communications in Mathematical Physics 57 (1977), 173 – 185, DOI: 10.1007/BF01625776.

K.J. Dykema and D. Shlyakhtenko, Exactness of Cuntz-Pimsner C∗-algebras, Proceedings of the Edinburgh Mathematical Society 44(2) (2001), 425 – 444, DOI: 10.1017/S001309159900125X.

S. Eilenberg, Automata, Languages and Machines, 1st edition, Vol. 59A, Academic Press, New York, 450 pages (1974).

A.A. Hassan, Semi generalized open sets and generalized semi closed sets in topological spaces, International Journal of Analysis and Applications 18(6) (2020), 1029 – 1036, DOI: 10.28924/2291-8639-18-2020-1029.

A.A. Hassan and M. Jawed, A commutative and compact derivations for W∗ algebra, International Journal of Analysis and Applications 18(4) (2020), 644 – 662, DOI: 10.28924/2291-8639-18-2020-644.

J.H. Hong, A. Skalski and W. Szymanski, On invariant MASAs for endomorphisms of the Cuntz algebras, Indiana University Mathematics Journal 59(6) (2010), 1873 – 1892, URL: https://www.jstor.org/stable/24903380.

A.A. Huef and I. Raeburn, The ideal structure of Cuntz-Krieger algebras, Ergodic Theory and Dynamical Systems 17(3) (1997), 611 – 624, DOI: 10.1017/S0143385797079200.

M. Izumi, Subalgebras of infinite C∗-algebras with finite Watatani indices I: Cuntz algebras, Communications in Mathematical Physics 155 (1993), 157 – 182, DOI: 10.1007/BF02100056.

M. Izumi, Subalgebras of infinite C∗-algebras with finite Watatani indices, II: Cuntz-Krieger algebras, Duke Mathematical Journal 91 (1998), 409 – 461, DOI: 10.1215/S0012-7094-98-09118-9.

V.F.R. Jones, On a family of almost commuting endomorphisms, Journal of Functional Analysis 122(1) (1994), 84 – 90, DOI: 10.1006/jfan.1994.1062.

Y. Katayama and H. Takehana, On automorphisms of generalized Cuntz algebras, International Journal of Mathematics 9(4) (1998), 493 – 512, DOI: 10.1142/S0129167X9800021X.

K. Kawamura, Polynomial endomorphisms of the Cuntz algebras arising from permutations: I – General theory, Letters in Mathematical Physics 71 (2005), 149 – 158, DOI: 10.1007/s11005-005-0344-8.

A. Kumjian, D. Pask and I. Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific Journal of Mathematics 184(1) (1998), 161 – 174, URL: https://msp.org/pjm/1998/184-1/pjm-v184-n1-p08-s.pdf.

K. Matsumoto and J. Tomiyama, Outer automorphisms on Cuntz algebras, Bulletin of the London Mathematical Society 25(1) (1993), 64 – 66, DOI: 10.1112/blms/25.1.64.

V.V. Nekrashevych, Cuntz-Pimsner algebras of group actions, Journal of Operator Theory 52(2) (2004), 223 – 249, URL: https://www.theta.ro/jot/archive/2004-052-002/2004-052-002-001.html.

D. Pask and A. Rennie, The noncommutative geometry of graph C∗-algebras I: The index theorem, Journal of Functional Analysis 233(1) (2006), 92 – 134, DOI: 10.1016/j.jfa.2005.07.009.

W. Szymanski, On localized automorphisms of the Cuntz algebras which preserve the diagonal subalgebra, in: New Development of Operator Algebras, RIMS Kôkyûroku 1587 (2008), 109 – 115, URL: https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1587-08.pdf.

J. Zacharias, Quasi-free automorphisms of Cuntz-Krieger-Pimsner algebras, in: C∗-Algebras, J. Cuntz and S. Echterhoff (editors), Springer, Berlin — Heidelberg (2000), DOI: 10.1007/978-3-642-57288-3_15.




How to Cite

Hassan, A. A. (2024). Localized Automorphisms and Endomorphisms. Communications in Mathematics and Applications, 14(5), 1575–1584. https://doi.org/10.26713/cma.v14i5.1971



Research Article