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Visualizing Automorphisms of Graph Algebras

Part of: Selfadjoint operator algebras Topological dynamics

Published online by Cambridge University Press:  01 February 2018

James Emil Avery ,
Rune Johansen  and
Wojciech Szymański
James Emil Avery*
Affiliation:
Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 København Ø, Denmark (avery@nbi.ku.dk)
Rune Johansen
Affiliation:
Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100 København Ø, Denmark (rune@cyx.dk)
Wojciech Szymański
Affiliation:
Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark (szymanski@imada.sdu.dk)
*
*Corresponding author.
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Abstract

Many fundamental properties of graph C*-algebras may be determined directly from the structure of the underlying graph, and because of this, they have been celebrated as C*-algebras that can be seen. This paper shows how permutative endomorphisms of graph C*-algebras can be represented by labelled directed multigraphs that give visual representations of the endomorphisms and facilitate computations. This formalism provides a useful calculus for permutative automorphisms and allows efficient exhaustive construction of such automorphisms.

Keywords

graph algebrapermutative endomorphismautomorphismtextile system

MSC classification

Primary: 46L40: Automorphisms
Secondary: 37B10: Symbolic dynamics
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 1 , February 2018 , pp. 215 - 249
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1. Abrams, G. and Aranda Pino, G., The Leavitt path algebra of a graph, J. Algebra 293(2) (2005), 319–334.CrossRefGoogle Scholar
2. Ara, P., Moreno, M. A. and Pardo, E., Nonstable K-theory for graph algebras, Algebr. Represent. Theory 10(2) (2007), 157–178.Google Scholar
3. Avery, J. E. and Johansen, R., Efficient computations with automorphisms of shift spaces, in preparation.Google Scholar
4. Avery, J. E., Johansen, R. and Simonsen, J. G., Invertibility of one-block codes, in preparation.Google Scholar
5. Bratteli, O. and Jorgensen, P. E. T., Iterated function systems and permutation representations of the Cuntz algebra, Mem. Amer. Math. Soc. 139(663) (1999), x+89.Google Scholar
6. Conti, R., Hong, J. H. and Szymański, W., Endomorphisms of graph algebras, J. Funct. Anal. 263(9) (2012), 2529–2554.Google Scholar
7. Conti, R., Hong, J. H. and Szymański, W., The restricted Weyl group of the Cuntz algebra and shift endomorphisms, J. Reine Angew. Math. 667 (2012), 177–191.Google Scholar
8. Conti, R., Kimberley, J. and Szymański, W., More localized automorphisms of the Cuntz algebras, Proc. Edinb. Math. Soc. (2) 53(3) (2010), 619–631.CrossRefGoogle Scholar
9. Conti, R. and Pinzari, C., Remarks on the index of endomorphisms of Cuntz algebras, J. Funct. Anal. 142(2) (1996), 369–405.Google Scholar
10. Conti, R., Rørdam, M. and Szymański, W., Endomorphisms of 𝒪 n which preserve the canonical UHF-subalgebra, J. Funct. Anal. 259(3) (2010), 602–617.Google Scholar
11. Conti, R. and Szymański, W., Computing the Jones index of quadratic permutation endomorphisms of 𝒪2 , J. Math. Phys. 50(1) (2009), 012705.Google Scholar
12. Conti, R. and Szymański, W., Labeled trees and localized automorphisms of the Cuntz algebras, Trans. Amer. Math. Soc. 363(11) (2011), 5847–5870.CrossRefGoogle Scholar
13. Conti, R. and Szymański, W., Automorphisms of the Cuntz algebras, in Progress in operator algebras, noncommutative geometry, and their applications, Proceedings of the 4th Annual Meeting of the European Noncommutative Geometry Network, Bucharest, 2011, pp. 1–15 (Theta, Bucharest, 2012).Google Scholar
14. Cuntz, J., Automorphisms of certain simple C* -algebras, in Quantum fields—algebras, processes Proceedings of the Symposium Bielefeld Encounters in Physics and Mathematics, Volume II, pp. 187–196 (Springer, Vienna, 1980).CrossRefGoogle Scholar
15. Cuntz, J. and Krieger, W., A class of C* -algebras and topological Markov chains, Invent. Math. 56(3) (1980), 251–268.Google Scholar
16. Fowler, N. J., Laca, M. and Raeburn, I., The C*-algebras of infinite graphs, Proc. Amer. Math. Soc. 128(8) (2000), 2319–2327.Google Scholar
17. The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.7.7. (2015).Google Scholar
18. Hayashi, T., On normalizers of C*-subalgebras in the Cuntz algebra 𝒪 n , J. Operator Theory 69(2) (2013), 525–533.Google Scholar
19. Hopenwasser, A., Peters, J. R. and Power, S. C., Subalgebras of graph C*-algebras, New York J. Math. 11 (2005), 351–386 (electronic).Google Scholar
20. Izumi, M., Subalgebras of infinite C*-algebras with finite Watatani indices. I. Cuntz algebras, Comm. Math. Phys. 155(1) (1993), 157–182.Google Scholar
21. Izumi, M., Subalgebras of infinite C*-algebras with finite Watatani indices. II. Cuntz-Krieger algebras, Duke Math. J. 91(3) (1998), 409–461.Google Scholar
22. Kawamura, K., Polynomial endomorphisms of the Cuntz algebras arising from permutations. I. General theory, Lett. Math. Phys. 71(2) (2005), 149–158.Google Scholar
23. Kumjian, A., Pask, D. and Raeburn, I., Cuntz-Krieger algebras of directed graphs, Pacific J. Math. 184(1) (1998), 161–174.Google Scholar
24. Kumjian, A., Pask, D., Raeburn, I. and Renault, J., Graphs, groupoids, and Cuntz-Krieger algebras, J. Funct. Anal. 144(2) (1997), 505–541.CrossRefGoogle Scholar
25. Lind, D. and Marcus, B., An introduction to symbolic dynamics and coding (Cambridge University Press, Cambridge, 1995).CrossRefGoogle Scholar
26. Longo, R., A duality for Hopf algebras and for subfactors. I, Comm. Math. Phys. 159(1) (1994), 133–150.CrossRefGoogle Scholar
27. Nagy, G. and Reznikoff, S., Abelian core of graph algebras, J. Lond. Math. Soc. (2) 85(3) (2012), 889–908.Google Scholar
28. Nasu, M., Textile systems for endomorphisms and automorphisms of the shift, Mem. Amer. Math. Soc. 114(546) (1995), viii+215.Google Scholar
29. Raeburn, I., Graph algebras, CBMS Regional Conference Series in Mathematics, Volume 103 (CBMS, Washington, DC, 2005).Google Scholar
30. Skalski, A., Noncommutative topological entropy of endomorphisms of Cuntz algebras II, Publ. Res. Inst. Math. Sci. 47(4) (2011), 887–896.Google Scholar
31. Skalski, A. and Zacharias, J., Noncommutative topological entropy of endomorphisms of Cuntz algebras, Lett. Math. Phys. 86(2–3) (2008), 115–134.Google Scholar
32. Tomforde, M., Leavitt path algebras with coefficients in a commutative ring, J. Pure Appl. Algebra 215(4) (2011), 471–484.Google Scholar
33. Webster, S. B. G., The path space of a directed graph, Proc. Amer. Math. Soc. 142(1) (2014), 213–225.Google Scholar