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Tytuł artykułu

Free probability induced by electric resistance networks on energy Hilbert spaces

Treść / Zawartość
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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We show that a class of countable weighted graphs arising in the study of electric resistance networks (ERNs) are naturally associated with groupoids. Starting with a fixed ERN, it is known that there is a canonical energy form and a derived energy Hilbert space Hε. From Hε, one then studies resistance metrics and boundaries of the ERNs. But in earlier research, there does not appear to be a natural algebra of bounded operators acting on Hε. With the use of our ERN-groupoid, we show that Hε may be derived as a representation Hilbert space of a universal representation of a groupoid algebra [formula], and we display other representations. Among our applications, we identify a free structure of [formula] in terms of the energy form.
Rocznik
Strony
549--598
Opis fizyczny
Bibliogr. 33 poz.
Twórcy
autor
  • St. Ambrose University Department of Mathematics 518 W. Locust St., Davenport, Iowa, 52803, USA, chowoo@sau.edu
Bibliografia
  • [1] I. Cho, Graph von Neumann algebras, ACTA Appl. Math. 95 (2007), 95–134.
  • [2] I. Cho, Graph Groupoids and Partial Isometries, LAP Publisher, 2009, ISBN: 978-3-8383-1397-9.
  • [3] I. Cho, P.E.T. Jorgensen, C∗-subalgebras generated by partial isometries, JMP, DOI: 10.1063/1.3056588, (2009).
  • [4] I. Cho, P.E.T. Jorgensen, C∗-subalgebras generated by a single operator in B(H), ACTA Appl. Math. 108 (2009), 625–664.
  • [5] I. Cho, P.E.T. Jorgensen, Directed graphs, von Neumann algebras, and index, Alg. Rep. Theo. (2010), DOI: 10.1007/s10488-010-9233-7.
  • [6] P.E.T. Jorgensen, E. Pearse, Operator theory of electric resistance networks, (2009) Preprint.
  • [7] P.E.T. Jorgensen, E.P.J. Pearse, Multiplication operators on the energy space, arXive: 1007.3516, (2010) Preprint.
  • [8] P.E.T. Jorgensen, E.P.J. Pearse, Resistance boundaries of infinite networks, arXive: 0909.1518, (2009) Preprint.
  • [9] P.E.T. Jorgensen, E.P.J. Pearse, A Hilbert space approach to effective resistance metric, arXive: 0906.2535, (2009) Preprint.
  • [10] P.E.T. Jorgensen, E.P.J. Pearse, A discrete Gauss-Green identity for unbounded Laplacian operators and transience of random walks, arXiv:0906.1586, (2009) Preprint.
  • [11] P.E.T. Jorgensen, A sampling theory for infinite weighted graphs, Opuscula Math. 31 (2011) 2, 209–236.
  • [12] S. Zuyev, Stochastic Geometry and Telecommunications Networks, New Perspectives in Stochastic Geometry, Oxford Publ. (2010), 520–554, MR: 2654689.
  • [13] A.M. Kazun, R. Szwarc, Jacobi matrices on trees, Colloq. Math. 118 (2010) 2, 465–497.
  • [14] M.D. Penrose, A.R. Wade, Random Directed and On-line Networks, New Perspectives in Stochastic Geometry, Oxford Publ. (2010), 248–274, MR: 2654689.
  • [15] M. Mesbahi, M. Egerstedt, Graph Theoretic Methods in Multiagent Networks, Princeton Ser. Appl. Math., Princeton Univ. Press, 2010, ISBN: 978-0-691-14061-2.
  • [16] B. Boyle, K. Cekala, D. Ferrone, N. Rifkin, A. Teplyaev, Electrical resistance of N-gasket fractal networks, Pacific J. Math. 233 (2007) 1, 15–40.
  • [17] E. Lopez, S. Carmi, S. Havlin, S.V. Buldyrev, H.E. Stanley, Anomalous electrical and frictionless flow conductance in complex networks, Phys. D. 224 (2006), 69–76.
  • [18] W.J. Tzeng, F.Y. Wu, Theory of Impedance Networks: The two-point impedance and LC resonances, J. Phys. A. 39 (2006) 27, 8579–8591.
  • [19] P. Boivin, J. Renault, Haussdorf-Young inequality for measured groupoids, von Neumann algebras in Sibiu, Theta Ser. Adv. Math. 10 (2008), 9–19.
  • [20] M. Amini, Tannaka-Krein duality for compact groupoids, Adv. Math. 214 (2007) 1, 78–91.
  • [21] H. Amiri, B.M. Lashkarizadeh, Square integrable representation of groupoids, ACTA Math. Sin. (Eng. Ser.) 23 (2007) 2, 327–340.
  • [22] G. Goehle, The Mackey machine for crossed products by regular groupoids I, Houston J. Math. 36 (2010) 2, 567–590.
  • [23] E.K. Magnani, K. Kangni, On a groupoid Gelfand pair, JP J. Alg. Number Theo. Appl. 16 (2010) 2, 109–118.
  • [24] M. Cucuringu, R.S. Strichartz, Infinitesimal resistance metrics on Sierpinski gasket type fractals, Anal. International Math. J. Anal. Appl. 28 (2008) 3, 319–331.
  • [25] E. Bendito, A. Carmona, A. Encinas, Equilibrium Measure, Poisson Kernel and Effective Resistance on Networks, Random Walks and Geometry, Walter de Gruyter Co. KG. Berlin, MR: 2087789, 2004.
  • [26] T. Hattori, A. Kasue, Functions of finite Dirichlet sums and compactifications of infinite graphs, Probabilistic Approach to Geometry, Adv. Stud. Pure Math. 57 (2010), 141–153.
  • [27] D. Yang, X. Gao, D-saturated property of the Cayley graphs of semigroups, Semigroup Forum 80 (2010) 1, 174–180.
  • [28] Z. Dvovak, B. Mohar, Spectral radius of finite and infinite planar graphs and of graphs of bounded genus, Euro. Conf. on Combinatorics, Graph Theory, Applications, Electron. Notes Discrete Math. 34 (2009), 101–105.
  • [29] P.G. Doyle, J.L. Snell, Random Walks and Electric Networks, The Carus Math. Mongraphs no. 22, (1984) MAA, ISBN: 0-88 385-024-9.
  • [30] R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, AMS Mem 132 (1998) 627.
  • [31] M. Bouzefrane, M. Chellali, A Note on global alliances in trees, Opuscula Math. 31 (2011) 2, 153–158.
  • [32] M.R. Buneci, Convolution algebras for topological groupoids with locally compact fibres, Opuscula Math. 31 (2011) 2, 159–172.
  • [33] P. Kurasov, M. Nowaczyk, Geometric properties of quantum graphs and vertex scattering matrices, Opuscula Math. 30 (2010) 3, 295-309.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-AGHS-0003-0005
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