Skip to main content

The semicircular law of free probability as noncommutative multivariable operator theory

Abstract

In this paper, we study semicircular elements induced by connected finite directed graphs. It is shown that if the graph groupoid \(\mathbb {G}\) of a given graph G contains at least one loop finite path, then it induces a semicircular element under suitable representations of \(\mathbb {G}\). As application, if a graph G is fractal (or, satisfies the fractal property) in a certain sense, then it automatically generates infinitely many semicircular elements.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Alpay, D., Jorgensen, P.E.T., Salomon, G.: On free stochastic processes and their derivatives. Stoch. Process. Appl. 124(10), 3392–3411 (2014)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Arveson, W.: Subalgebras of \(C^{*}\)-algebras. (III). Multivariable operator theory. Acta Math. 181(2), 159–228 (1998)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bartholdi, L., Grigorchuk, R., Nekrashevych, V.: From fractal groups to fractal sets. In: Grabner P., Woess W. (eds.) Fractals in Graz 2001. Analysis, dynamics, geometry, stochastics. Proceedings of the conference, Graz, Austria, June 2001. Trends in Mathematics, pp. 25–118. Basel: Birkhäuser (2003)

  4. 4.

    Berkolaiko, G., Kuchment, P.: Introduction to Quantum Graphs, Mathematical Surveys and Monograph, vol. 186. American Mathematical Society, Providence, RI (2013) (ISBN: 978-0-8218-9211-4)

  5. 5.

    Cho, I.: Graph von Neumann algebras. Acta Appl. Math. 95, 95–135 (2007)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Cho, I.: Frames, fractals and radial operators in Hilbert space. J. Math. Sci. Adv. Appl. 5(2), 333–393 (2010)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Cho, I.: Histories distorted by partial isometries. J. Math. Phys. 3 (2011) (article ID: P110301)

  8. 8.

    Cho, I.: Fractal properties in \(B(H)\) induced by partial isometries. Complex Anal. Oper. Theory 5(1), 1–40 (2011)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Cho, I.: Semicircular-like, and semicircular laws on Banach \(*\)-probability spaces induced by dynamical systems of the finite Adele Ring. Adv. Oper. Theory 4(1), 24–70 (2019)

  10. 10.

    Cho, I.: Free semicircular families in free product Banach \(*\)-algebras induced by \(p\)-Adic number fields \(\mathbb{Q}_{p}\) over primes \(p\). Complex Anal. Oper. Theory 11(3), 507–565 (2017)

  11. 11.

    Cho, I.: Banach-space operators acting on semicircular elements induced by orthogonal projections. Complex Anal. Oper. Theory 13(8), 4065–4115 (2019)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Cho, I., Jorgensen, P.E.T.: \(C^{*}\)-subalgebras generated by partial isometries. J. Appl. Math. Comput. 26, 1–48 (2008)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Cho, I., Jorgensen, P.E.T.: \(C^{*}\)-subalgebras generated by a single operator in \(B(H)\). Acta Appl. Math. 108, 625–664 (2009)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Cho, I., Jorgensen, P.E.T.: Deformations of semicircular and circular laws via \(p\)-Adic number fields \(\mathbb{Q}_{p}\) and sampling of primes. Opusc. Math. 39(6), 771–811 (2019)

    Google Scholar 

  15. 15.

    Cho, I., Jorgensen, P.E.T.: Certain \(*\)-homomorphisms on \(C^{*}\)-algebras and sequences of semicircular elements: a Banach space view. Ill. J. Math. 64(4), 519–567 (2020)

  16. 16.

    Connes, A.: Noncommutative Geometry, Lect. Note in Math., Math. Research Today and Tomorrow (Barcelona), vol. 1525, pp. 40–58. Springer, New York (1992) (MR:1247054)

  17. 17.

    Curto, R.: Two-Variable Weighted Shifts in Multivariable Operator Theory, Handbook of Analytic Operator Theory, pp. 17–63. CRP Press, Boca Raton (2019)

  18. 18.

    Dicks, W., Ventura, E.: The Group Fixed by a Family of Injective Endomorphisms of a Free Group, Contemp. Math., vol. 195. AMS, Providence, RI (1996)

  19. 19.

    Dutkay, D.E., Jorgensen, P.E.T.: Iterated Function Systems, Ruelle Operators and Invariant Projective Measures, Math. Comput. 75(256), 1931–1970 (2006)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Exel, R.: A new Look at the crossed-product of a \(C^{*}\)-algebra by a Semigroup of Endomorphisms. Ergodic Theory Dyn. Syst. 28(3), 749–789 (2008)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Farsi, C., Gillaspy, E., Jorgensen, P.E.T., Kang, S.: Purely atomic representations of higher-rank graph \(C^{*}\)-algebras. Integr. Equations Oper. Theory 90(6), 26 (2018) (Paper No. 67)

  22. 22.

    Farsi, C., Gillaspy, E., Jorgensen, P.E.T., Kang, S.: Monic representations of finite higher-rank graphs. Ergod. Theory Dyn. Syst. 40(5), 1238–1267 (2020)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Gill, A.: Introduction to the Theory of Finite-State Machines, vol. MR0209083. McGraw-Hill Book Co., New York (1962)

    MATH  Google Scholar 

  24. 24.

    Greene, D.C.V.: Free resolutions in multivariable operator theory. J. Funct. Anal. 200(2), 429–450 (2003)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Ghosh, A., Boyd, S., Saberi, A.: Minimizing effective resistance of a graph. SIAM. Rev. 50(1), 37–66 (2008)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Gibbons, A., Novak, L.: Hybrid Graph Theory and Network Analysis. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  27. 27.

    Gliman, R., Shpilrain, V., Myasnikov, A.G. (eds.): Computational and Statistical Group Theory, Contemporary Math, vol. 298. American Mathematical Society, Providence, RI (2001)

  28. 28.

    Hiai, F., Petz, D.: Law, The Semicircular, Variables, Free Random, Entropy, Math, Surveys and Monographs, vol. 77. American Mathematical Society, Providence RI (2000) (ISBN:0-8218-2081-8)

  29. 29.

    Jorgensen, P.E.T.: Use of operator algebras in the analysis of measures from wavelets and iterated function systems. In: Han D et al. (eds.) Operator theory, operator algebras, and applications. Proceedings of the 25th Great Plains Operator Theory Symposium, University of Central Florida, FL, USA, June 7–12, 2005. Contemporary Mathematics 414, pp. 13–26. Providence, RI: American Mathematical Society (AMS) (2006)

  30. 30.

    Jorgensen, P.E.T., Pearse, E.P.J.: Resistance boundaries of infinite networks, random walks, boundaries and spectra. Progr. Probab. 64, 111–142 (2011)

    MATH  Google Scholar 

  31. 31.

    Jorgensen, P.E.T., Pearse, E.P.J.: Continuum vs. discrete networks, graph Laplacians, and reproducing Kernel Hilbert spaces. J. Math. Anal. Appl. 469(2), 765–807 (2019)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Jorgensen, P.E.T., Song, M.: Entropy encoding, Hilbert spaces, and Kahunen–Loeve transforms. J. Math. Phys. 48(10), 103503 (2007)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Jorgensen, P.E.T., Schmitt, L.M., Werner, R.F.: \(q\)-Canonical commutation relations and stability of the Cuntz algebra. Pac. J. Math. 165(1), 131–151 (1994)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Kribs, D.W.: On bilateral shifts in noncommutative multivariable operator theory. Indiana Univ. Math. J. 52(6), 1595–1614 (2003)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Kribs, D.W.: Quantum causal histories and the directed graph operator framework (2005). arXiv:math.OA/0501087v1 (preprint)

  36. 36.

    Kribs, D.W., Jury, M.T.: Ideal structure in free semigroupoid algebras from directed graphs. J. Oper. Theory 53(2), 273–302 (2005)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Kucherenko, I.V.: On the structurization of a class of reversible cellular automata. Diskret. Mat. 19(3), 102–121 (2007)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Kigami, J., Strichartz, R.S., Walker, K.C.: Constructing a Laplacian on the diamond fractal. Exp. Math. 10(3), 437–448 (2001)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Kostrykin, V., Potthoff, J., Schrader, R.: Construction of the paths of Brownian motions on star graphs I. Commun. Stoch. Anal. 6(2), 223–245 (2012)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Lind, D.A.: Entropies of automorphisms of a topological Markov shift. Proc. AMS 99(3), 589–595 (1987)

    MathSciNet  Article  Google Scholar 

  41. 41.

    Lind, D.A., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995)

  42. 42.

    Marshall, C.W.: Applied Graph Theory. Wiley, New York (1971) (ISBN: 0-471-57300-0)

  43. 43.

    Mitchener, P.D.: \(C^{*}\)-Categories, groupoid actions, equivalent KK-theory, and the Baum–Connes conjecture. J. Funct. Anal. 214(1), 1–39 (2004)

    MathSciNet  Article  Google Scholar 

  44. 44.

    Myasnikov, A.G., Shapilrain, V. (eds.): Group Theory, Statistics and Cryptography, Contemporary Math., vol. 360. AMS, Providence, RI (2003)

  45. 45.

    Popescu, G.: Noncommutative multivariable operator theory. Integr. Equations Oper. Theory 75(1), 87–133 (2013)

    MathSciNet  Article  Google Scholar 

  46. 46.

    Potgieter, P.: Nonstandard analysis, fractal properties and Brownian motion. Fractals 17(1), 117–129 (2009)

    MathSciNet  Article  Google Scholar 

  47. 47.

    Radulescu, F.: Random matrices, amalgamated free products and subfactors of the \(C^{*}\)-algebra of a free group, of noninteger index. Invent. Math. 115, 347–389 (1994)

    MathSciNet  Article  Google Scholar 

  48. 48.

    Raeburn, I.: Graph Algebras, CBMS No. 3. AMS, Providence, RI (2005)

  49. 49.

    Scapellato, R., Lauri, J.: Topics in Graph Automorphisms and Reconstruction, London Math. Soc., Student Text, vol. 54. Cambridge University Press, Cambridge (2003)

  50. 50.

    Schiff, J.L.: Cellular Automata, Discrete View of the World, Wiley-Interscience Series in Disc. Math. & Optimazation. Wiley, New York (2008) (ISBN: 978-0-470-16879-0)

  51. 51.

    Shirai, T.: The spectrum of infinite regular line graphs. Trans. AMS. 352(1), 115–132 (2000)

    MathSciNet  Article  Google Scholar 

  52. 52.

    Solel, B.: You can see the arrows in a Quiver Operator Algebras. J. Aust. Math. Soc. 77(1), 111–122 (2004)

    MathSciNet  Article  Google Scholar 

  53. 53.

    Somoulya, J., Mandal, A.: Quantum symmetry of graph \(C^{*}\)-algebras at critical inverse temperature. Studia Math. 256(1), 1–20 (2021)

    MathSciNet  Article  Google Scholar 

  54. 54.

    Speicher, R.: Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory, vol. 132, no. 627. Memoirs of the AMS, Providence, RI (1998)

  55. 55.

    Vega, V.: Finite Directed Graphs and \(W^{*}\)-Correspondences, Ph.D. thesis, University of Iowa (2007)

  56. 56.

    Voiculescu, D., Dykema, K., Nica, A.: Free Random Variables, vol. 1. CRM Monograph Series. Published by Amer. Math. Soc (1992). ISBN:978-0-8218-1140-5

  57. 57.

    Weintraub, S.H.: Representation Theory of Finite Groups: Algebra and Arithmetic, Grad. Studies in Math, vol. 59. AMS, Providence RI (2003)

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ilwoo Cho.

Additional information

Communicated by M. S. Moslehian.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cho, I., Jorgensen, P.E.T. The semicircular law of free probability as noncommutative multivariable operator theory. Adv. Oper. Theory 6, 69 (2021). https://doi.org/10.1007/s43036-021-00163-0

Download citation

Keywords

  • Graphs
  • Graph groupoids
  • Semicircular elements
  • The semicircular law

Mathematics Subject Classification

  • 47A99
  • 05C62
  • 17A50
  • 18B40