Skip to main content
SpringerLink
Log in

Applications of Automata and Graphs: Labeling-Operators in Hilbert Space I

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

We show that certain representations of graphs by operators on Hilbert space have uses in signal processing and in symbolic dynamics. Our main result is that graphs built on automata have fractal characteristics. We make this precise with the use of Representation Theory and of Spectral Theory of a certain family of Hecke operators. Let G be a directed graph. We begin by building the graph groupoid \(\Bbb{G}\) induced by G, and representations of  \(\Bbb{G}\) . Our main application is to the groupoids defined from automata. By assigning weights to the edges of a fixed graph G, we give conditions for \(\Bbb{G}\) to acquire fractal-like properties, and hence we can have fractaloids or G-fractals. Our standing assumption on G is that it is locally finite and connected, and our labeling of G is determined by the “out-degrees of vertices”. From our labeling, we arrive at a family of Hecke-type operators whose spectrum is computed. As applications, we are able to build representations by operators on Hilbert spaces (including the Hecke operators); and we further show that automata built on a finite alphabet generate fractaloids. Our Hecke-type operators, or labeling operators, come from an amalgamated free probability construction, and we compute the corresponding amalgamated free moments. We show that the free moments are completely determined by certain scalar-valued functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Balacheff, F.: Volume entropy, systole and stable norm on graphs. Preprint arXiv:math.MG/0411578v1 (2004)

  2. Bartholdi, L., Grigorchuk, R., Nekrashevych, V.: From fractal groups to fractal sets. Preprint arXiv:math.GR/0202001v4 (2002)

  3. Bell, G.C.: Growth of the asymptotic dimension function for groups. Preprint (2005)

  4. Cho, I.: The moments of certain perturbed operators of the radial operator of the free group factor L(F N ). JAA 5(3), 137–165 (2007)

    MATH  Google Scholar 

  5. Cho, I.: Characterization of free blocks of a right graph von Neumann algebra. To be appeared in Complex Anal. Oper. Theory (2007)

  6. Cho, I.: Direct producted W *-probability spaces and corresponding free stochastic integration. Bull. Korean Math. Soc. 44(1), 131–150 (2007)

    MathSciNet  MATH  Google Scholar 

  7. Cho, I.: Vertex-compressed algebras of a graph von Neumann algebra. Submitted to ACTA. Appl. Math. (2007)

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Cho, I.: Labeling operators of graph groupoids. Preprint (2008)

  10. Cho, I.: Group-freeness and certain amalgamated freeness. J. Korean Math. Soc. 45(3), 597–609 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cho, I., Jorgensen, P.E.T.: C *-subalgebras generated by partial isometries in B(H). Submitted to JMP (2007)

  12. Cho, I., Jorgensen, P.: C *-algebras generated by partial isometries. To appear in JAMC (2008)

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

  14. Dutkay, D.E., Jorgensen, P.E.T.: Iterated function systems, Ruelle operators and invariant projective measures. Preprint arXiv:math.DS/0501077/v3 (2005)

  15. Exel, R.: A new look at the crossed-product of a C *-algebra by a semigroup of endomorphisms. Preprint (2005)

  16. Fannes, M., Nachtergaele, B., Werner, R.F.: Finitely correlated states on quantum spin chains. Commun. Math. Phys. 144(3), 443–490 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fannes, M., Nachtergaele, B., Werner, R.F.: Entropy estimates for finitely correlated states. Ann. Inst. H. Poincare Phys. Theor. 57(3), 259–277 (1992)

    MathSciNet  MATH  Google Scholar 

  18. Fannes, M., Nachtergaele, B., Werner, R.F.: Ground states of VSB-models on Cayley trees. J. Stat. Phys. 66, 939–973 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fannes, M., Nachtergaele, B., Werner, R.F.: Finitely correlated Pure states. J. Funct. Anal. 120(2), 511–534 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gibbons, A., Novak, L.: Hybrid Graph Theory and Network Analysis. Cambridge Univ. Press, Cambridge (1999). ISBN:0-521-46117-0

    Google Scholar 

  21. Gill, A.: Introduction to the Theory of Finite-State Machines. McGraw-Hill, New York (1962). MR0209083 (34\#8891)

    MATH  Google Scholar 

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

    Google Scholar 

  23. Guido, D., Isola, T., Lapidus, M.L.: A trace on fractal graphs and the ihara zeta function. Preprint arXiv:math.OA/0608060v1 (2006)

  24. Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Language, and Computation. Addision-Wesley, Reading (1979). ISBN:0-201-02988-X

    Google Scholar 

  25. Jorgensen, P.E.T.: Use of operator algebras in the analysis of measures from wavelets and iterated function systems. Preprint (2005)

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

  27. 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  Google Scholar 

  28. Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebra. Grad. Stud. Math., vol. 15. AMS, Providence (1997)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

  31. Kribs, D.W., Jury, M.T.: Ideal structure in free semigroupoid algebras from directed graphs. Preprint

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  35. Lind, D.A., Schmidt, K.: Symbolic and algebraic dynamical systems. In: Handbook of Dynamical System, vol. 1A, pp. 765–812. Elsevier, Amsterdam (2002)

    Chapter  Google Scholar 

  36. Lind, D.A., Tuncel, S.: A spanning tree invariant for Markov shifts. IMA Vol. Math. Appl. 123, 487–497 (2001)

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  38. Mitchener, P.D.: C *-categories, groupoid actions, equivalent KK-theory, and the Baum-Connes conjecture. Preprint arXiv:math.KT/0204291v1 (2005)

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

    Google Scholar 

  40. Potgieter, P.: Nonstandard analysis, fractal properties and Brownian motion. Preprint arXiv:math.FA/0701649v1 (2007)

  41. Powers, R.T.: Heisenberg model and a random walk on the permutation group. Lett. Math. Phys. 1(2), 125–130 (1975)

    Article  MathSciNet  Google Scholar 

  42. Powers, R.T.: Resistance inequalities for KMS-states of the isotropic Heisenberg model. Commun. Math. Phys. 51(2), 151–156 (1976)

    Article  MathSciNet  Google Scholar 

  43. Powers, R.T.: Registance inequalities for the isotropic Heisenberg ferromagnet. J. Math. Phys. 17(10), 1910–1918 (1976)

    Article  MathSciNet  Google Scholar 

  44. Radulescu, F.: Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index. Invent. Math. 115, 347–389 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  45. Raeburn, I.: Graph Algebras. CBMS, vol. 3. AMS, Providence (2005)

    MATH  Google Scholar 

  46. Renault, J.: A Groupoid Approach to C *-algebras. In: Lect. Notes in Math., vol. 793. Springer, Berlin (1980). ISBN:3-540-09977-8

    MATH  Google Scholar 

  47. Sakai, S.: C *-Algebras and W *-Algebras. Springer, Berlin (1971). MR number: MR0442701

    Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  51. Solel, B.: You can see the arrows in a Quiver operator algebras. Preprint (2000)

  52. Speicher, R. (eds.): Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. AMS Mem. 132(627) (1998)

  53. Thompson, S., Cho, I.: Powers of mutinomials in commutative algebras. (Undergraduate Research) Submitted to PMEJ (2008)

  54. Thompson, S., Mendoza, C.M., Kwiatkowski, A.J., Cho, I.: Lattice paths satisfying the axis property. (Undergraduate Research) Preprint (2008)

  55. Vega, V.: Finite directed graphs and W *-correspondences. Ph.D. thesis, Univ. of Iowa (2007)

  56. Voiculescu, D.: Symmetries of some reduced free product C *-algebras. In: Lect. Notes in Math., vol. 1132, pp. 556–588. Springer, Berlin (1985)

    Google Scholar 

  57. Voiculescu, D., Dykemma, K., Nica, A.: Free Random Variables. CRM Monograph Series, vol 1 (1992)

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

    MATH  Google Scholar 

  59. Wigner, E.P.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. 62(2), 548–564 (1955)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dep. of Math., Saint Ambrose Univ., 421 Ambrose Hall, 518 W. Locust St., Davenport, IA, 52803, USA

    Ilwoo Cho & Palle E. T. Jorgensen

  2. Dep. of Math., Univ. of Iowa, 25 McLean Hall, Iowa City, IA, 52242, USA

    Ilwoo Cho & Palle E. T. Jorgensen

Authors
  1. Ilwoo Cho
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Palle E. T. Jorgensen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ilwoo Cho.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cho, I., Jorgensen, P.E.T. Applications of Automata and Graphs: Labeling-Operators in Hilbert Space I. Acta Appl Math 107, 237–291 (2009). https://doi.org/10.1007/s10440-008-9380-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-008-9380-6

Keywords

Mathematics Subject Classification (2000)

Search

Navigation