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BRANCHING SYSTEMS FOR HIGHER-RANK GRAPH C*-ALGEBRAS

Part of: Ergodic theory Selfadjoint operator algebras

Published online by Cambridge University Press:  12 March 2018

DANIEL GONÇALVES ,
HUI LI  and
DANILO ROYER
DANIEL GONÇALVES
Affiliation:
Departamento de Matemática, Universidade Federal de Santa Catarina, Florianópolis 88040-900, Brazil e-mail: [email protected]
HUI LI
Affiliation:
Research Center for Operator Algebras and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, Department of Mathematics, East China Normal University, 3663 Zhongshan North Road, Putuo District, Shanghai 200062, China e-mail: [email protected]
DANILO ROYER
Affiliation:
Departamento de Matemática, Universidade Federal de Santa Catarina, Florianópolis 88040-900, Brazil e-mail: [email protected]
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Abstract

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We define branching systems for finitely aligned higher-rank graphs. From these, we construct concrete representations of higher-rank graph C*-algebras on Hilbert spaces. We prove a generalized Cuntz–Krieger uniqueness theorem for periodic single-vertex 2-graphs. We use this result to give a sufficient condition under which representations of periodic single-vertex 2-graph C*-algebras arising from branching systems are faithful.

MSC classification

Primary: 46L05: General theory of $C^*$-algebras
Secondary: 37A55: Relations with the theory of $C^*$-algebras
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 3 , September 2018 , pp. 731 - 751
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

References

REFERENCES

1. Blackadar, B., Operator algebras: Theory of C*-algebras and von Neumann algebras, Operator algebras and non-commutative geometry, III (Springer-Verlag, Berlin, 2006), xx+517p.Google Scholar
2. Bratteli, O. and Jorgensen, P. E. T., Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale N, Integral Equ. Oper. Theory 28 (1997), 382–443.CrossRefGoogle Scholar
3. Bratteli, O. and Jorgensen, P. E. T., Iterated function systems and permutation representations of the Cuntz algebra, Mem. Am. Math. Soc. 139 (1999), x+89.Google Scholar
4. Brown, J. H., Nagy, G. and Reznikoff, S., A generalized Cuntz–Krieger uniqueness theorem for higher-rank graphs, J. Funct. Anal. 266 (2014), 2590–2609.Google Scholar
5. Carlsen, T. M., Kang, S., Shotwell, J. and Sims, A., The primitive ideals of the Cuntz–Krieger algebra of a row-finite higher-rank graph with no sources, J. Funct. Anal. 266 (2014), 2570–2589.Google Scholar
6. Chen, X. W., Irreducible representations of Leavitt path algebras, Forum Math. 27 (2015), 549–574.CrossRefGoogle Scholar
7. Davidson, K. R. and Yang, D., Periodicity in rank 2 graph algebras, Canad. J. Math. 61(6) (2009), 1239–1261.Google Scholar
8. Davidson, K. R. and Yang, D., Representations of higher rank graph algebras, NY J. Math. 15 (2009), 169–198.Google Scholar
9. Devulder, A., The speed of a branching system of random walks in random environment, Statist. Probab. Lett. 77 (2007), 1712–1721.CrossRefGoogle Scholar
10. Farsi, C., Gillaspy, E., Kang, S. and Packer, J., Separable representations, KMS states, and wavelets for higher-rank graphs, J. Math. Anal. Appl. 434 (2016), 241–270.CrossRefGoogle Scholar
11. Farsi, C., Gillaspy, E., Kang, S. and Packer, J., Wavelets and graph C*-algebras, (2016). arXiv:1601.00061v1.Google Scholar
12. Gonçalves, D., Li, H. and Royer, D., Branching systems and general Cuntz–Krieger uniqueness theorem for ultragraph C*-algebras, Int. J. Math. 27 (10) (2016), 1650083 (26 pg).Google Scholar
13. Gonçalves, D., Li, H. and Royer, D., Faithful representations of graph algebras via branching systems, Can. Math. Bull. 59 (2016), 95–103.CrossRefGoogle Scholar
14. Gonçalves, D. and Royer, D., Branching systems and representations of Cohn–Leavitt path algebras of separated graphs, J. Algebra 422 (2015), 413–426.CrossRefGoogle Scholar
15. Gonçalves, D. and Royer, D., Graph C*-algebras, branching systems and the Perron–Frobenius operator, J. Math. Anal. Appl. 391 (2012), 457–465.CrossRefGoogle Scholar
16. Gonçalves, D. and Royer, D., On the representations of Leavitt path algebras, J. Algebra 333 (2011), 258–272.Google Scholar
17. Gonçalves, D. and Royer, D., Perron–Frobenius operators and representations of the Cuntz–Krieger algebras for infinite matrices, J. Math. Anal. Appl. 351 (2009), 811–818.Google Scholar
18. Gonçalves, D. and Royer, D., Unitary equivalence of representations of algebras associated with graphs, and branching systems, Funct. Anal. Appl. 45 (2011), 45–59.Google Scholar
19. Hazrat, R. and Rangaswamy, K. M., On graded irreducible representations of Leavitt path algebras, J. Algebra 450 (2016), 458–486.Google Scholar
20. Hochberg, K. J. and Greven, A., On the use of the Laplace functional for two-level branching systems, Int. J. Pure Appl. Math. 55 (2009), 165–172.Google Scholar
21. Huef, A., Laca, M., Raeburn, I. and Sims, A., KMS states on the C*-algebra of a higher-rank graph and periodicity in the path space, J. Funct. Anal. 268 (2015), 1840–1875.CrossRefGoogle Scholar
22. Katsura, T., The ideal structures of crossed products of Cuntz algebras by quasi-free actions of abelian groups, Can. J. Math. 55 (2003), 1302–1338.Google Scholar
23. Kumjian, A. and Pask, D., Higher rank graph C*-algebras, NY J. Math. 6 (2000), 1–20.Google Scholar
24. Kumjian, A., Pask, D., Sims, A. and Whittaker, M. F., Topological spaces associated to higher-rank graphs, J. Comb. Theory Ser. A 143 (2016), 19–41.Google Scholar
25. Marcolli, M. and Paolucci, A. M., Cuntz–Krieger algebras and wavelets on fractals, Complex Anal. Oper. Theory 5 (2011), 41–81.CrossRefGoogle Scholar
26. Raeburn, I., Sims, A. and Yeend, T., Higher-rank graphs and their C*-algebras, Proc. Edinb. Math. Soc. 46 (2003), 99–115.Google Scholar
27. Raeburn, I., Sims, A. and Yeend, T., The C*-algebras of finitely aligned higher-rank graphs, J. Funct. Anal. 213 (2004), 206–240.CrossRefGoogle Scholar
28. Robertson, G. and Steger, T., Affine buildings, tiling systems and higher rank Cuntz–Krieger algebras, J. R. Angew. Math. 513 (1999) 115-144.Google Scholar
29. Royden, H.L., Real analysis, 2nd edition (The Macmillan Co. Collier-Macmillan Ltd., London), xii+349p.Google Scholar
30. Salinier, B. and Strandh, R., Efficient simulation of forward-branching systems with constructor systems, J. Symb. Comput. 22 (1996), 381–399.Google Scholar
31. Szymański, W., General Cuntz–Krieger uniqueness theorem, Int. J. Math. 13 (2002), 549–555.Google Scholar
32. Yang, D., Periodic k-graph algebras revisited, J. Aust. Math. Soc. 99 (2015), 267–286.Google Scholar