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Non-Commutative Methods for the K-Theory of C*-Algebras of Aperiodic Patterns from Cut-and-Project Systems

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Abstract

We investigate the C*-algebras associated to aperiodic structures called model sets obtained by the cut-and-project method. These C*-algebras are Morita equivalent to crossed product C*-algebras obtained from dynamics on a disconnected version of the internal space. This construction may be made from more general data, which we call a hyperplane system. From a hyperplane system, others may be constructed by a process of reduction and we show how the C*-algebras involved are related to each other. In particular, there are natural elements in the Kasparov KK-groups for the C*-algebra of a hyperplane system and that of its reduction. The induced map on K-theory fits in a six-term exact sequence. This provides a new method of the computation of the K-theory of such C*-algebras which is done completely in the setting of non-commutative geometry.

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References

  1. Bellissard, J.: K-theory of C*-algebras in solid state physics. In: Statistical Mechanics and Field Theory, Mathematical Aspects, Dorlas, T.C., Hugenholz, M.N., Winnink, M. (eds.), Lecture Notes in Physics, 257, Berlin-New York: Springer, 1986, pp. 99–156

  2. Bellissard, J.: Noncommutative geometry for aperiodic solids. In: Geometric and Topological Methods for Quantum Field Theory (Villa de Leyva, 2001), River Edge, NJ: World Sci. Publishing, 2003, pp. 86–156

  3. Bellissard, J., Contensou, E., Legrand, A.: K-theorie des quasi-crystaux, image par la trace: le cas du reseau octoganol, C.R. Acad. Sci. (Paris), t. 327, Serie I, 197–200 (1998)

  4. Bellissard, J., Herrmann, D., Zarrouati, M.: Hull of aperiodic solids and gap labelling theorems. In: Directions in Mathematical Quasicrystals, Baake, M.B., Moody, R.V. (eds.), CRM Monograph Series, Volume 13, Providence, RI: Amer. Math. Soc., 2000, pp. 207–259

  5. Bellissard J., Iochum B., Testard D.: Continuity properties of electronic spectrum of 1D quasicrystals. Commun. Math. Phys. 141, 353–380 (1991)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. Bellissard J., Savinien J.: A spectral sequence for the K-theory of tiling spaces. Eng. Th. Dyn. Syst. 29, 997–1031 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cuntz, J.: Generalized homomorphisms between C*-algebras and KK-theory. In: Dynamics and Processes (Bielefeld, 1981), Blanchard, P., Streit, L. (eds.), Lecture Notes in Math. 1031, Berlin: Springer-Verlag, 1983, pp. 31–45

  8. Duneau M., Katz A.: Quasiperiodic patterns. Phys. Rev. Lett. 54, 2688–2691 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  9. Duneau M., Katz A.: Quasiperiodic patterns and icosehedral symmetry. J. Phys (France) 47, 181–196 (1986)

    MathSciNet  Google Scholar 

  10. Elser V.: Indexing problems in quasicrystal diffraction. Phys. Rev. B 32, 4892–4898 (1985)

    Article  ADS  Google Scholar 

  11. Elser V.: Comment on “Quasicrystals: a new class of ordered structures”. Phys. Rev. Lett. 54, 1730 (1985)

    Article  ADS  Google Scholar 

  12. Forrest A., Hunton J.: The cohomology and K-theory of commuting homeomorphisms of the Cantor set. Erg. Th. Dyn. Syst. 19, 611–625 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  13. Forrest, A., Hunton, J., Kellendonk, J.: Topological invariants for projection method patterns. Mem. A.M.S. 159 (758) (2002)

  14. Grünbaum B., Shephard G.C.: Tilings and Patterns. Freeman, New York (1987)

    MATH  Google Scholar 

  15. Hibbert, F., Gratias, D. (eds): Lectures on Quasicrystals. Editions de Physique, Les Ulis (1994)

    Google Scholar 

  16. Janot C.: Quasicrystals. A Primer. Clarendon, Oxford (1992)

    Google Scholar 

  17. Kalugin P.A., Kitaev A.Y., Levitov L.S.: Al 0.86 MN 0.14: a six-dimensional crystal. JETP Lett. 41, 145–149 (1985)

    ADS  Google Scholar 

  18. Kalugin P.A., Kitaev A.Y., Levitov L.S.: 6-dimensional properties of Al 0.86 MN 0.14. J. Phys. Lett. (France) 46, L601–L607 (1985)

    Google Scholar 

  19. Kellendonk, J., Putnam, I.F.: Tilings, C*-algebras and K-theory. In: Directions in Mathematical Quasicrystals, Baake, M., Moody, R.V. (eds.), CRM Monograph Series, Volume 13, Providence, RI: Amer. Math. Soc., 2000, pp. 177–206

  20. Kramer, P., Neri, R.: On periodic and non-periodic space fillings of E m obtained by projections. Acta. Cryst. A 40, 580–587 (1984), and (Erratum) Acta. Cryst. A 41, 619 (1985)

  21. Le, T.T.Q.: Local rules for aperiodic tilings. In: The Mathematics of Long Range Aperiodic Order, Moody, R.V. ed., Dordrecht: Kluwer, 1996, pp. 331–366

  22. Levine D., Steinhradt P.J.: Quasicrystals: a new class of ordered structures. Phys. Rev. Lett. 53, 2477–2480 (1984)

    Article  ADS  Google Scholar 

  23. Meyer, Y.: Algebraic Numbers and Harmonic Analysis. Amsterdam: North-Holland, 1972

  24. Meyer, Y.: Quasicrystals, Diophantine approximation and algebraic numbers. In: Beyond Quasicrystals, eds. Axel, F., Gratias, D., Berlin: Springer-Verlag, 1995, pp. 3–16

  25. Moody, R.V.: Model sets: a survey. In: From Quasicrystals to More Complex Systems, eds. F. Axel, F. Dnoyer, J.P. Gazeau, Centre de physique Les Houches, Berlin-Heidelberg-New York: Springer-Verlag, 2000

  26. Muhly P.S., Renault J.N., Williams D.P.: Equivalence and isomorphism for groupoid C*-algebras. J. Op. Th. 17, 3–22 (1987)

    MATH  MathSciNet  Google Scholar 

  27. Pedersen, G.K.: Analysis NOW. Graduate Texts in Mathematics, Vol. 118, Berlin-Heidelberg-New York: Springer-Verlag, 1988

  28. Putnam I.F.: On the K-theory of C*-algebras of principal groupoids. Rocky Mountain J. Math. 28, 1483–1518 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  29. Putnam I.F., Schmidt K., Skau C.F.: C*-algebras associated with Denjoy homeomorphisms of the circle. J. Op. Th. 16, 99–126 (1986)

    MATH  MathSciNet  Google Scholar 

  30. Rieffel M.A.: Applications of strong Morita equivalence to transformation group C*-algebras. Proc. Symp. Pure Math. 38, 299–310 (1982)

    MathSciNet  Google Scholar 

  31. Schechtman D., Blech I., Gratias D., Cahn J.W.: Metallic Phase with Long-Range Orientational Order and No Translational Symmetry. Phys. Rev. Lett 51, 1951–1953 (1984)

    Article  ADS  Google Scholar 

  32. Senechal M.: Quasicrystals and Discrete Geometry. Cambridge University Press, Cambridge (1995)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Victoria, Victoria, B.C., Canada, V8W 3R4

    Ian F. Putnam

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  1. Ian F. Putnam
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Correspondence to Ian F. Putnam.

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Communicated by A. Connes

Supported in part by a grant from NSERC, Canada.

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Putnam, I.F. Non-Commutative Methods for the K-Theory of C*-Algebras of Aperiodic Patterns from Cut-and-Project Systems. Commun. Math. Phys. 294, 703–729 (2010). https://doi.org/10.1007/s00220-009-0968-0

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