Labeled trees and localized automorphisms of the Cuntz algebras

Conti, Roberto; Szymański, Wojciech

doi:10.1090/S0002-9947-2011-05234-7

Labeled trees and localized automorphisms of the Cuntz algebras
HTML articles powered by AMS MathViewer

by Roberto Conti and Wojciech Szymański

Trans. Amer. Math. Soc. 363 (2011), 5847-5870

DOI: https://doi.org/10.1090/S0002-9947-2011-05234-7

Published electronically: June 2, 2011

PDF | Request permission

Abstract:

We initiate a detailed and systematic study of automorphisms of the Cuntz algebras $\mathcal {O}_n$ which preserve both the diagonal and the core UHF-subalgebra. A general criterion of invertibility of endomorphisms yielding such automorphisms is given. Combinatorial investigations of endomorphisms related to permutation matrices are presented. Key objects entering this analysis are labeled rooted trees equipped with additional data. Our analysis provides insight into the structure of $\textrm {Aut}(\mathcal {O}_n)$ and leads to numerous new examples. In particular, we completely classify all such automorphisms of $\mathcal {O}_2$ for the permutation unitaries in $\otimes ^4 M_2$. We show that the subgroup of $\textrm {Out}(\mathcal {O}_2)$ generated by these automorphisms contains a copy of the infinite dihedral group ${\mathbb Z} \rtimes {\mathbb Z}_2$.

References

R. J. Archbold, On the “flip-flop” automorphism of $C^{\ast } (S_{1},\,S_{2})$, Quart. J. Math. Oxford Ser. (2) 30 (1979), no. 118, 129–132. MR 534827, DOI 10.1093/qmath/30.2.129
Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
Roberto Conti and Francesco Fidaleo, Braided endomorphisms of Cuntz algebras, Math. Scand. 87 (2000), no. 1, 93–114. MR 1776967, DOI 10.7146/math.scand.a-14301
Roberto Conti and Claudia Pinzari, Remarks on the index of endomorphisms of Cuntz algebras, J. Funct. Anal. 142 (1996), no. 2, 369–405. MR 1423039, DOI 10.1006/jfan.1996.0154
Joachim Cuntz, Simple $C^*$-algebras generated by isometries, Comm. Math. Phys. 57 (1977), no. 2, 173–185. MR 467330
J. Cuntz, Automorphisms of certain simple $C^{\ast }$-algebras, Quantum fields—algebras, processes (Proc. Sympos., Univ. Bielefeld, Bielefeld, 1978) Springer, Vienna, 1980, pp. 187–196. MR 601811
Joachim Cuntz and Wolfgang Krieger, A class of $C^{\ast }$-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268. MR 561974, DOI 10.1007/BF01390048
Rolf Gohm, A probabilistic index for completely positive maps and an application, J. Operator Theory 54 (2005), no. 2, 339–361. MR 2186358
Alan Hopenwasser, Justin R. Peters, and Stephen C. Power, Subalgebras of graph $C^*$-algebras, New York J. Math. 11 (2005), 351–386. MR 2188247
Masaki Izumi, Subalgebras of infinite $C^*$-algebras with finite Watatani indices. I. Cuntz algebras, Comm. Math. Phys. 155 (1993), no. 1, 157–182. MR 1228532
Masaki Izumi, Finite group actions on $C^*$-algebras with the Rohlin property. I, Duke Math. J. 122 (2004), no. 2, 233–280. MR 2053753, DOI 10.1215/S0012-7094-04-12221-3
V. F. R. Jones, On a family of almost commuting endomorphisms, J. Funct. Anal. 122 (1994), no. 1, 84–90. MR 1274584, DOI 10.1006/jfan.1994.1062
V. Jones and V. S. Sunder, Introduction to subfactors, London Mathematical Society Lecture Note Series, vol. 234, Cambridge University Press, Cambridge, 1997. MR 1473221, DOI 10.1017/CBO9780511566219
Katsunori Kawamura, Polynomial endomorphisms of the Cuntz algebras arising from permutations. I. General theory, Lett. Math. Phys. 71 (2005), no. 2, 149–158. MR 2134694, DOI 10.1007/s11005-005-0344-8
Katsunori Kawamura, Branching laws for polynomial endomorphisms of Cuntz algebras arising from permutations, Lett. Math. Phys. 77 (2006), no. 2, 111–126. MR 2251300, DOI 10.1007/s11005-006-0050-1
Akitaka Kishimoto and Alexander Kumjian, Crossed products of Cuntz algebras by quasi-free automorphisms, Operator algebras and their applications (Waterloo, ON, 1994/1995) Fields Inst. Commun., vol. 13, Amer. Math. Soc., Providence, RI, 1997, pp. 173–192. MR 1424962
Roberto Longo, A duality for Hopf algebras and for subfactors. I, Comm. Math. Phys. 159 (1994), no. 1, 133–150. MR 1257245
Kengo Matsumoto, Orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras, Pacific J. Math. 246 (2010), no. 1, 199–225. MR 2645883, DOI 10.2140/pjm.2010.246.199
Kengo Matsumoto and Jun Tomiyama, Outer automorphisms on Cuntz algebras, Bull. London Math. Soc. 25 (1993), no. 1, 64–66. MR 1190366, DOI 10.1112/blms/25.1.64
Hiroki Matui, Classification of outer actions of $\Bbb Z^N$ on $\scr O_2$, Adv. Math. 217 (2008), no. 6, 2872–2896. MR 2397470, DOI 10.1016/j.aim.2007.11.023
S. C. Power, Homology for operator algebras. III. Partial isometry homotopy and triangular algebras, New York J. Math. 4 (1998), 35–56. MR 1609023
S. C. Power, Subalgebras of graph $C^*$-algebras, Lecture Notes for the Summer School Course at WOAT 2006, Lisbon, 1–5 September 2006.
Iain Raeburn, Graph algebras, CBMS Regional Conference Series in Mathematics, vol. 103, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2005. MR 2135030, DOI 10.1090/cbms/103
Mikael Rørdam, A short proof of Elliott’s theorem: ${\scr O}_2\otimes {\scr O}_2\cong {\scr O}_2$, C. R. Math. Rep. Acad. Sci. Canada 16 (1994), no. 1, 31–36. MR 1276341
John Spielberg, Free-product groups, Cuntz-Krieger algebras, and covariant maps, Internat. J. Math. 2 (1991), no. 4, 457–476. MR 1113572, DOI 10.1142/S0129167X91000260
W. Szymański, On localized automorphisms of the Cuntz algebras which preserve the diagonal subalgebra, in ‘New Development of Operator Algebras’, R.I.M.S. Kôkyûroku 1587 (2008), 109–115.