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Unitary equivalence of representations of graph algebras and branching systems. (English. Russian original) Zbl 1271.46043

Funct. Anal. Appl. 45, No. 2, 117-127 (2011); translation from Funkts. Anal. Prilozh. 45, No. 2, 45-59 (2011).
Summary: It is shown that, for many countable graphs, every representation of the associated graph algebra in a separable Hilbert space is unitarily equivalent to a representation obtained via branching systems.

MSC:

46L05 General theory of \(C^*\)-algebras
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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References:

[1] M. Abe and K. Kawamura, ”Recursive fermion system in Cuntz algebra. I. Embeddings of fermion algebra into Cuntz algebra,” Comm. Math. Phys., 228:1 (2002), 85–101. · Zbl 1029.46079 · doi:10.1007/s002200200651
[2] M. Abe and K. Kawamura, ”Pseudo Cuntz algebra and recursive FP ghost system in string theory,” Internat. J. Modern. Phys., A18:4 (2003), 607–625. · Zbl 1091.81503 · doi:10.1142/S0217751X0301379X
[3] T. Bojdecki, L. G. Gorostiza, and A. Talarczyk, ”Limit theorems for occupation time fluctuations of branching systems. I. Long-range dependence,” Stochastic Process. Appl., 116:1 (2006), 1–18. · Zbl 1082.60024 · doi:10.1016/j.spa.2005.07.002
[4] O. Bratteli and P. E. T. Jorgensen, ”A connection between multiresolution wavelet theory of scale N and representations of the Cuntz algebra ON,” in: Operator Algebras and Quantum Field Theory (Rome, 1996), Int. Press, Cambridge, MA, 1997, 151–163.
[5] A. Devulder, ”The speed of a branching system of random walks in random environment,” Statist. Probab. Lett., 77:18 (2007), 1712–1721. · Zbl 1138.60341 · doi:10.1016/j.spl.2007.04.010
[6] G. Gierz, ”Representation of spaces of compact operators and applications to the approximation property,” Arch. Math., 30:1 (1978), 622–628. · Zbl 0391.46059 · doi:10.1007/BF01226110
[7] D. Gonçalves and D. Royer, ”Graph algebras as subalgebras of the bounded operators in L 2(\(\mathbb{R}\)),” submitted to publication; http://arxiv.org/abs/0908.1055v1 .
[8] D. Gonçalves and D. Royer, ”Perron-Frobenius operators and representations of the Cuntz-Krieger algebras for infinite matrices,” J. Math. Anal. Appl., 351 (2009), 811–818. · Zbl 1160.46033 · doi:10.1016/j.jmaa.2008.11.018
[9] D. Gonçalves and D. Royer, ”On the representations of Leavitt path algebras,” J. Algebra, 333 (2011), 258–272. · Zbl 1235.16014 · doi:10.1016/j.jalgebra.2011.02.034
[10] K. J. Hochberg and A. Greven, ”On the use of the Laplace functional for two-level branching systems,” Int. J. Pure Appl. Math., 55:2 (2009), 165–172. · Zbl 1180.60078
[11] T. Katsura, A. Sims, and M. Tomforde, ”Realization of AF -algebras as graph algebras, Exel-Laca algebras, and ultragraph algebras,” J. Funct. Anal., 257:5 (2009), 1589–1620. · Zbl 1186.46058 · doi:10.1016/j.jfa.2009.05.002
[12] K. Kawamura, Representations of the Cuntz algebra O2 arising from complex quadratic transformations–Annular basis of L2(C), preprint RIMS, 2003, No. 1418.
[13] K. Kawamura, Three representations of the Cuntz algebra O2 by a pair of operators arising from a Z2-graded dynamical system, Preprint RIMS, 2003, No. 1415.
[14] K. Kawamura and O. Suzuki, Construction of orthonormal basis on self-similar sets by generalized permutative representations of the Cuntz algebras, Preprint RIMS, 2003, No. 1408.
[15] J. A. López-Mimbela and A. Murillo-Salas, ”Laws of large numbers for the occupation time of an age-dependent critical binary branching system,” ALEA Lat. Am. J. Probab. Math. Stat., 6 (2009), 115–131. · Zbl 1162.60337
[16] N. J. Fowler, M. Laca, and I. Raeburn, ”The C*-algebras of infinite graphs,” Proc. Amer. Math. Soc., 128:8 (2000), 2319–2327. · Zbl 0956.46035 · doi:10.1090/S0002-9939-99-05378-2
[17] T. Poomsa-ard, ”Identities in universal graph algebras,” Int. Math. Forum, 4:33–36 (2009), 1715–1722.
[18] B. Salinier and R. Strandh, ”Efficient simulation of forward-branching systems with constructor systems,” J. Symbolic Comput., 22:4 (1996), 381–399. · Zbl 0869.68061 · doi:10.1006/jsco.1996.0058
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