Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 064, 46 pages      arXiv:1401.4622      https://doi.org/10.3842/SIGMA.2014.064
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

Non-Commutative Resistance Networks

Marc A. Rieffel
Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA

Received January 22, 2014, in final form June 10, 2014; Published online June 14, 2014

Abstract
In the setting of finite-dimensional $C^*$-algebras ${\mathcal A}$ we define what we call a Riemannian metric for ${\mathcal A}$, which when ${\mathcal A}$ is commutative is very closely related to a finite resistance network. We explore the relationship with Dirichlet forms and corresponding seminorms that are Markov and Leibniz, with corresponding matricial structure and metric on the state space. We also examine associated Laplace and Dirac operators, quotient energy seminorms, resistance distance, and the relationship with standard deviation.

Key words: resistance network; Riemannian metric; Dirichlet form; Markov; Leibniz seminorm; Laplace operator; resistance distance; standard deviation.

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References

  1. Albeverio S., Høegh-Krohn R., Dirichlet forms and Markov semigroups on $C^*$-algebras, Comm. Math. Phys. 56 (1977), 173-187.
  2. Beggs E.J., Majid S., $*$-compatible connections in noncommutative Riemannian geometry, J. Geom. Phys. 61 (2011), 95-124, arXiv:0904.0539.
  3. Beurling A., Deny J., Espaces de Dirichlet. I. Le cas élémentaire, Acta Math. 99 (1958), 203-224.
  4. Blackadar B., Operator algebras. Theory of $C^{*}$-algebras and von Neumann algebras, Encyclopaedia of Mathematical Sciences, Vol. 122, Springer-Verlag, Berlin, 2006.
  5. Cipriani F., Dirichlet forms and Markovian semigroups on standard forms of von Neumann algebras, J. Funct. Anal. 147 (1997), 259-300.
  6. Cipriani F., Dirichlet forms on noncommutative spaces, in Quantum Potential Theory, Lecture Notes in Math., Vol. 1954, Springer, 2008, 161-276.
  7. Cipriani F., Sauvageot J.L., Derivations as square roots of Dirichlet forms, J. Funct. Anal. 201 (2003), 78-120.
  8. Connes A., Noncommutative geometry, Academic Press, Inc., 1994.
  9. Davies E.B., Analysis on graphs and noncommutative geometry, J. Funct. Anal. 111 (1993), 398-430.
  10. Davies E.B., Lindsay J.M., Noncommutative symmetric Markov semigroups, Math. Z. 210 (1992), 379-411.
  11. Evans D.E., Conditionally completely positive maps on operator algebras, Quart. J. Math. Oxford Ser. (2) 28 (1977), 271-283.
  12. Fröhlich J., Grandjean O., Recknagel A., Supersymmetric quantum theory and non-commutative geometry, Comm. Math. Phys. 203 (1999), 119-184, math-ph/9807006.
  13. Gracia-Bondía J.M., Várilly J.C., Figueroa H., Elements of noncommutative geometry, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Boston, Inc., Boston, MA, 2001.
  14. Halbout G., Tang X., Dunkl operator and quantization of ${\mathbb Z}_2$-singularity, J. Reine Angew. Math. 673 (2012), 209-235, arXiv:0908.4301.
  15. Ivanciuc O., QSAR and QSPR molecular descriptors computed from the resistance distance and electrical conductance matrices, Models Chem. 137 (2000), 607-631.
  16. Jorgensen P.E.T., Pearse E.P.J., Operator theory of electrical resistance networks, arXiv:0806.3881.
  17. Jorgensen P.E.T., Pearse E.P.J., A Hilbert space approach to effective resistance metric, Complex Anal. Oper. Theory 4 (2010), 975-1013, arXiv:0906.2535.
  18. Junge M., Mei T., Parcet J., An invitation to harmonic analysis associated with semigroups of operators, in Proceedings of 9th International Conference "Harmonic Analysis and PDE's'', to appear, arXiv:1304.4922.
  19. Kadison R.V., Ringrose J.R., Fundamentals of the theory of operator algebras. Vol. II. Advanced theory, Graduate Studies in Mathematics, Vol. 16, Amer. Math. Soc., Providence, RI, 1997.
  20. Kaliszewski S., Patani N., Quigg J., Characterizing graph $C^*$-correspondences, Houston J. Math. 38 (2012), 751-759, arXiv:0906.3311.
  21. Kigami J., Analysis on fractals, Cambridge Tracts in Mathematics, Vol. 143, Cambridge University Press, Cambridge, 2001.
  22. Klein D.J., Palacios J.L., Randic M., Trinajstic N., Random walks and chemical graph theory, J. Chem Inf. Comput. Sci. 44 (2004), 1521-1525.
  23. Klein D.J., Randić M., Resistance distance, J. Math. Chem. 12 (1993), 81-95.
  24. Lindblad G., On the generators of quantum dynamical semigroups, Comm. Math. Phys. 48 (1976), 119-130.
  25. Majid S., Noncommutative Riemannian geometry on graphs, J. Geom. Phys. 69 (2013), 74-93, arXiv:1011.5898.
  26. McRae B.H., Isolation by resistance, Evolution 60 (2006), 1551-1561.
  27. Peterson J., A 1-cohomology characterization of property (T) in von Neumann algebras, Pacific J. Math. 243 (2009), 181-199, math.OA/0409527.
  28. Preskill J., Lecture notes on quantum computation, 1997, available at http://www.theory.caltech.edu/people/preskill/ph229/notes/chap3.pdf.
  29. Raeburn I., Williams D.P., Morita equivalence and continuous-trace $C^*$-algebras, Mathematical Surveys and Monographs, Vol. 60, Amer. Math. Soc., Providence, RI, 1998.
  30. Rieffel M.A., Induced representations of $C^{\ast}$-algebras, Adv. Math. 13 (1974), 176-257.
  31. Rieffel M.A., Metrics on states from actions of compact groups, Doc. Math. 3 (1998), 215-229, math.OA/9807084.
  32. Rieffel M.A., Metrics on state spaces, Doc. Math. 4 (1999), 559-600, math.OA/9906151.
  33. Rieffel M.A., Matrix algebras converge to the sphere for quantum Gromov-Hausdorff distance, Mem. Amer. Math. Soc. 168 (2004), 67-91, math.OA/0108005.
  34. Rieffel M.A., Leibniz seminorms for "matrix algebras converge to the sphere'', in Quanta of Maths, Clay Math. Proc., Vol. 11, Amer. Math. Soc., Providence, RI, 2010, 543-578, arXiv:0707.3229.
  35. Rieffel M.A., Vector bundles and Gromov-Hausdorff distance, J. $K$-Theory 5 (2010), 39-103, math.MG/0608266.
  36. Rieffel M.A., Standard deviation is a strongly Leibniz seminorm, New York J. Math. 20 (2014), 35-56, arXiv:1208.4072.
  37. Rosenberg J., Levi-Civita's theorem for noncommutative tori, SIGMA 9 (2013), 071, 9 pages, arXiv:1307.3775.
  38. Ruan Z.J., Subspaces of $C^*$-algebras, J. Funct. Anal. 76 (1988), 217-230.
  39. Sauvageot J.L., Tangent bimodule and locality for dissipative operators on $C^*$-algebras, in Quantum Probability and Applications, IV (Rome, 1987), Lecture Notes in Math., Vol. 1396, Springer, Berlin, 1989, 322-338.
  40. Sauvageot J.L., Quantum Dirichlet forms, differential calculus and semigroups, in Quantum Probability and Applications, V (Heidelberg, 1988), Lecture Notes in Math., Vol. 1442, Springer, Berlin, 1990, 334-346.
  41. Sinayskiy I., Petruccione F., Efficiency of open quantum walk implementation of dissipative quantum computing algorithms, Quantum Inf. Process. 11 (2012), 1301-1309, arXiv:1401.6658.
  42. Takesaki M., Theory of operator algebras. I, Springer-Verlag, New York - Heidelberg, 1979.
  43. Weaver N., Lipschitz algebras, World Sci. Publ. Co., Inc., River Edge, NJ, 1999.
  44. Wiseman H.M., Milburn G.J., Quantum measurement and control, Cambridge University Press, Cambridge, 2010.
  45. Zhang H., Yang Y., Resistance distance and Kirchoff index in circulant graphs, Int. J. Quantum Chem. 107 (2007), 330-339.
  46. Zhao B., One noncommutative differential calculus coming from the inner derivation, math.OA/0611600.

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