#### Title

#### Document Type

Article

#### Abstract

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 × ℤ^{d} → Σ be a free and minimal action of ℤ^{d} on Σ with invariant probability measure *μ*. Let *μ*:*C*(Σ) → ℂ and *τ*_{μ}:*C*(Σ) ⋊ ℤ^{d} → ℂ be the traces induced by *μ* and denote likewise their induced maps in *K*-theory. Then *μ*(*K*_{0}(*C*(Σ))) = *τ*_{μ}(*K*_{0}(*C*(Σ) × ℤ^{d})) as subsets of ℝ.

We shall try to explain why this core result has anything to do with something called gap labeling.

#### Disciplines

Mathematics

#### Recommended Citation

C. Schochet, Gap Labeling, featured review in *Mathematical Reviews* 2005f:46121