The Gap Labelling Theorem: the case of automatic sequences Published in “Quantum and non-commutative analysis” (Kyoto, 1992), pp. 179-181, Math. Phys. Stud., 16, Kluwer Acad. Publ., Dordrecht, 1993. Jean BELLISSARD Universit´ e Paul Sabatier, Laboratoire de Physique Quantique 118, Route de Narbonne, 31062 Toulouse Cedex FRANCE Let H be a separable Hilbert space. Let G be a locally compact group acting on H by a projective unitary representation U . A selfadjoint operator H acting on H is called homogeneous with respect to the G action, if the family of its translated is precompact in the strong resolvent topology. The closure Ω of this set is called the “hull” of the pair (H,G), and G acts on it by homeomorphisms. Ω is a compact metrizable space. We will denote by H ω the self adjoint operator corresponding to ω ∈ Ω. It satisfies the covariance relation: U (g )H ω U (g ) -1 = H gω ω ∈ Ω, g ∈ G, and the map ω ∈ Ω → H ω is continuous for the strong resolvent topology. It follows that the spectrum of H ω is contained in the spectrum of H for all ω ∈ Ω. Let A be the C*-algebra generated in H by bounded functions of the H ω ’s. Given any pair Γ 1 , Γ 2 of gaps in the spectrum of H , the eigenprojection P (Γ 1 , Γ 2 ) corresponding to the part of the spectrum between these two gaps, defines an element of the countable abelian group K 0 (A). This is the “Abstract Gap Labelling Theorem”. The labelling is invariant under a norm-resolvent perturbation of H . Let us consider now the special case for which G is either R D or Z D , and H is L 2 (G). Let us assume in addition that H is bounded from below and that the resolvent (z - H ) -1 admits a distribution kernel G z (x,y) which decreases fast enough as | x - y |→ ∞. Then one can show that A is contained in the crossed-product C*-algebra C (Ω) × G. In addition, given an invariant probability measure P on Ω the trace per unit volume exists P-almost surely and defines a trace τ on A. The “integrated density of states” (IDS) of H is defined by: N (E )= lim |Λ|→∞ #{eigenvalues of H ω | Λ ≤ E } | Λ | , where | Λ | denotes the volume of Λ, and H ω | Λ denotes the restriction of the operator H ω to the (open) set Λ with some boundary condition. In most examples of interest, it can be shown that the limit exists and is independent of ω, P-almost surely. Moreover, the “Shubin formula” holds namely: N (E )= τ (χ(H ≤ E )), where χ(H ≤ E ) is the eigenprojection of H in the W*-algebra L ∞ (A,τ ) on the part of the spectrum contained in the interval (-∞,E ].