Bellissard's gap labeling conjecture for quasicrystals without magnetic field

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Let Λ\Lambda be the quasicrystal with transversal Ωtrans\Omega_{trans}, and let Aθ=0\mathcal{A}_{\theta=0} be its twisted groupoid C∗C^*-algebra for zero magnetic field. The trace induces the gap-labeling map

Tr∗:K0(Aθ=0)⟶R.Tr_*:K_0(\mathcal{A}_{\theta=0})\longrightarrow \mathbb{R}.

The set of gap labels is given by

∗∗Bellissard′sgaplabelingconjecture.∗∗Tr∗(K0(Aθ=0))=∫ΩtransC(Ωtrans,Z⁡),\mathbf{**Bellissard's gap labeling conjecture.**} \quad Tr_*(K_0(\mathcal{A}_{\theta=0}))=\int_{\Omega_{trans}} C(\Omega_{trans},\operatorname{\mathbb{Z}}),

which is precisely the group generated by the patch frequencies of Λ\Lambda. This conjecture predicts that, in the absence of a magnetic field, all gap labels are determined by the transversal and its patch frequencies. The source does not state whether the conjecture has been resolved.

References

Primary source

Michael Kreisel, “Gabor Frames for Quasicrystals, K-theory, and Twisted Gap Labeling”, arXiv:1411.7269 (2014).

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