This joint review covers:
Moulay-Tahar Benameur and Hervé Oyono-Oyono, Gap-labelling for quasi-crystals (proving a conjecture by J. Bellissard). (English summary) Operator algebras and mathematical physics, (Constanţa, 2001), 11-22, Theta, Bucharest, 2003.
Jerome Kaminker and Ian Putnam, A proof of the gap labeling conjecture, Michigan Mathematical Journal 51(3) (2003), 537-546.
Jean Bellisard, Riccardo Benedetti and Jean-Marc Gambaudo, Spaces of tilings, finite telescopic approximations, and gap-labeling, Communications in Mathematical Physics 261(1) (2006), 1-41.
The gap labeling theorem was originally conjectured by Bellissard [in From number theory to physics (Les Houches, 1989), 538-630, Springer, Berlin, 1992; MR1221111 (94e:46120)]. The problem arises in a mathematical version of solid state physics in the context of aperiodic tilings. Its three proofs, discovered independently by the authors above, all lie in K-theory. Here is the core result of these papers:
Let Σ be a Cantor set and let Σ × ℤd → Σ be a free and minimal action of ℤd on Σ with invariant probability measure μ. Let μ:C(Σ) → ℂ and τμ:C(Σ) ⋊ ℤd → ℂ be the traces induced by μ and denote likewise their induced maps in K-theory. Then μ(K0(C(Σ))) = τμ(K0(C(Σ) × ℤd)) as subsets of ℝ.
We shall try to explain why this core result has anything to do with something called gap labeling.
C. Schochet, Gap Labeling, featured review in Mathematical Reviews 2005f:46121