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We point out that quasicrystals actually occur in nature; see [45] for a survey. For the Schr¨odinger operator in the so-called tight binding representation for a quasicrystal whose atomic sites are given by an aperiodic tiling, the relevant C*-algebra is the C*-algebra of the tiling. See Sect. 4 of [4]. We have just seen that the C*-algebras associated with three different broad classes of tilings have real rank zero, that is, every selfadjoint element, including bounded continuous real functions of H , can be approximated arbitrarily closely in norm by selfadjoint elements with finite spectrum.

R(M(E,S),n ) are pairwise disjoint. Thus the M(E,S),k are graphs in G(E,S) with the required properties. 1 for s(K). 10. Let d be a positive integer, let X be the Cantor set, and let Zd act freely and minimally on X. 2, and Cr∗ (G) is simple. Proof. 9. 11. Let d be a positive integer, let X be the Cantor set, and let Zd act freely and minimally on X. Then: (1) C ∗ (Zd , X) has (topological) stable rank one in the sense of [42]. (2) C ∗ (Zd , X) has real rank zero in the sense of [10]. (3) Let p, q ∈ M∞ (C ∗ (Zd , X)) be projections such that τ (p) < τ (q) for all normalized traces τ on C ∗ (Zd , X).

Ergod. Th. Dynam. Sys. 18, 509–537 (1998) 50. : Talk at the conference “A periodic Order: Dynamical Systems, Combinatorics, and Operators”, Banff International Research Station, 29 May-3 June 2004 Communicated by Y. Kawahigashi Commun. Math. Phys. 1007/s00220-004-1281-6 Communications in Mathematical Physics Global Existence for the Einstein Vacuum Equations in Wave Coordinates Hans Lindblad1, , Igor Rodnianski2, 1 Mathematics Department, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0112, USA 2 Department Mathematics, Fine Hall, Princeton University, Princeton, NJ 08544-1000, USA Received: 11 December 2003 / Accepted: 28 September 2004 Published online: 8 March 2005 – © Springer-Verlag 2005 Abstract: We prove global stability of Minkowski space for the Einstein vacuum equations in harmonic (wave) coordinate gauge for the set of restricted data coinciding with the Schwarzschild solution in the neighborhood of space-like infinity.

### Communications in Mathematical Physics - Volume 256 by M. Aizenman (Chief Editor)

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