Quasicrystals, aperiodic order, and groupoid von Neumann algebras

Lenz, Daniel; Stollmann, Peter

Comptes Rendus. Mathématique, Tome 334 (2002) no. 12, pp. 1131-1136.

On introduit des opérateurs « tight binding » pour des quasicristaux paramétrés par des ensembles de Delone. On peut regarder ces opérateurs dans le contexte naturel des algèbres de von Neumann. Un tel point de vue permet d'étudier la théorie spectrale. En particulier la densité d'états intégrée est liée à une trace de l'algèbre.

We introduce tight binding operators for quasicrystals that are parametrized by Delone sets. These operators can be regarded in a natural operator algebra framework that encodes the long range aperiodic order. This algebraic point of view allows us to study spectral theoretic properties. In particular, the integrated density of states of the tight binding operators is related to a canonical trace on the associated von Neumann algebra.

[1] Avron, J.; Simon, B. Almost periodic Schrödinger operators, II: The integrated density of states, Duke Math. J., Volume 50 (1982), pp. 369-391

[2] Bellissard, J.; Lima, R.; Testard, D. Almost periodic Schrödinger operators, Mathematics + Physics, 1, World Scientific, Singapore, 1995, pp. 1-64

[3] Bellissard, J. K-theory of $C^{*}$-algebras in solid state physics, Statistical Mechanics and Field Theory: Mathematical Aspects, Groningen, 1985, Lecture Notes in Phys., 257, Springer, Berlin, 1986, pp. 99-156

[4] Bellissard, J. Gap labelling theorems for Schrödinger operators (Waldschmidt, M.; Moussa, P.; Luck, J.M.; Itzykson, C., eds.), From Number Theory to Physics, Springer, Berlin, 1992, pp. 539-630

[5] Bellissard, J.; Hermann, D.J.L.; Zarrouati, M. Hulls of aperiodic solids and gap labelling theorem, Directions in Mathematical Quasicrystals, CRM Monogr. Ser., 13, American Mathematical Society, Provicence, RI, 2000, pp. 207-258

[6] Carmona, R.; Lacroix, J. Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, 1990

[7] Coburn, L.A.; Moyer, R.D.; Singer, I.M. $C^{*}$-algebras of almost periodic pseudo-differential operators, Acta Math., Volume 130 (1973), pp. 279-307

[8] Connes, A. Sur la théorie non commutative de l'intégration, Lecture Notes in Math., 725, Springer, Berlin, 1979

[9] Geerse, C.P.M.; Hof, A. Lattice gas models on self-similar aperiodic tilings, Rev. Math. Phys., Volume 3 (1991), pp. 163-221

[10] Hof, A. A remark on Schrödinger operators on aperiodic tilings, J. Statist. Phys., Volume 81 (1996), pp. 851-855

[11] Hof, A. Some remarks on discrete aperiodic Schrödinger operators, J. Statist. Phys., Volume 72 (1993), pp. 1353-1374

[12] Janot, C. Quasicrystals: A Primer, Oxford University Press, Oxford, 1992

[13] Kellendonk, J. Noncommutative geometry of tilings and gap labelling, Rev. Math. Phys., Volume 7 (1995), pp. 1133-1180

[14] Kellendonk, J. The local structure of tilings and their integer grouip of coinvariants, Comm. Math. Phys., Volume 187 (1997), pp. 115-157

[15] Kellendonk, J.; Putnam, I.F. Tilings; $C^{*}$-algebras, and K-theory, Directions in Mathematical Quasicrystals, CRM Monogr. Ser., 13, American Mathematical Society, Providence, RI, 2000, pp. 177-206

[16] J.C. Lagarias, Geometric models for quasicrystals I. Delone sets of finite type, Ergodic Theory Dynamical Systems, to appear

[17] J.C. Lagarias, Geometric models for quasicrystals II. Local rules under isometries, Ergodic Theory Dynamical Systems, to appear

[18] J. Lagarias, P.A.B. Pleasants, Repetitive delone sets and quasicrystals, Ergodic Theory Dynamical Systems, to appear

[19] Lenz, D. Random operators and crossed products, Math. Phys. Anal. Geom., Volume 2 (1999), pp. 197-220

[20] D. Lenz, N. Peyerimhof, I. Veselic, Von Neumann algebras, groupoids and the integrated density of states, eprint: arXiv | arXiv

[21] D. Lenz, P. Stollmann, Delone dynamical systems and associated random operators, eprint: arXiv | arXiv

[22] D. Lenz, P. Stollmann, An ergodic theorem for Delone dynamical systems and existence of the density of states, in preparation

[23] Pastur, L.; Figotin, A. Spectra of Random and Almost Periodic Operators, Springer-Verlag, Berlin, 1992

[24] Schlottmann, M. Generalized model sets and dynamical systems (Baake, M.; Moody, R.V., eds.), Directions in Mathematical Quasicrystals, CRM Monogr. Ser., American Mathematical Society, Providence RI, 2000, pp. 143-159

[25] Senechal, M. Quasicrystals and Geometry, Cambridge University Press, Cambridge, 1995

[26] Shechtman, D.; Blech, I.; Gratias, D.; Cahn, J.W. Metallic phase with long-range orientational order and no translation symmetry, Phys. Rev. Lett., Volume 53 (1984), pp. 1951-1953

[27] Shubin, M. The spectral theory and the index of elliptic operators with almost periodic coefficients, Russian Math. Surveys, Volume 34 (1979)

[28] Solomyak, B. Dynamics of self-similar tilings, Ergodic Theory Dynamical Systems, Volume 17 (1997), pp. 695-738

[29] Solomyak, B. Spectrum of a dynamical system arising from Delone sets (Patera, J., ed.), Quasicrystals and Discrete Geometry, Fields Institute Monographs, 10, American Mathematical Society, Providence, RI, 1998, pp. 265-275

[30] Stollmann, P. Caught by Sisorder: Bound States in Random Media, Progress in Math. Phys., 20, Birkhäuser, Boston, 2001