×

Gabor frames for quasicrystals, \(K\)-theory, and twisted gap labeling. (English) Zbl 1360.19011

Quasicrystals \(\Lambda \subset \mathbb R^{2d}\) are generalizations of lattices which have enough lattice-like structure: are relatively dense (\(\rho(\Lambda) := \sup_{z\in \mathbb R^{2d}}\inf_{\lambda\in\Lambda} |z-\lambda|< \infty\)), unifomly discrete (up to translation there are only finitely many \(r\)-patches \(B_r(z) \cap \Lambda\) for all ball \(B_r(z)\) with radius \(r\)), have finite local complexity (FLC: there are only finitely many patches up to translation, for any real \(r\)) and have uniform cluster frequencies (UCF: there exist limits \(\mathrm{freq}(P,\Lambda) = \lim_{k\to \infty}\frac{L_P(B_{r_k}-z)}{vol(B_{r_k})}\) uniformly in \(z\) and independently of the choice of balls \(B_{r_k}\), where for each \(r\)-patch \(P\) and subset \(A\subset \mathbb R^{2d}\), \(L_P(A) = \{ z\in \mathbb R^{2d} | P-z \subset A \cap \Lambda \}\)). Two quasicrystals \(\lambda\) and \(\Lambda'\) are locally isomorphic iff any \(r\)-patch that appeared in \(\Lambda\) also appears as an \(r\)-patch in \(\Lambda\) and conversely. The hull \(\Omega_\Lambda\) of \(\Lambda\) is the equivalence class of all quasicrystals \(\lambda'\), locally isomorphic with \(\Lambda\). It is the closure of the orbit \(O_\Lambda= \{ \Lambda -z| z\in \mathbb R^{2d}\}\). The canonical transversal \(\Omega_{\mathrm{trans}}=\{ \Lambda' \in \Omega_\Lambda | 0 \in \Lambda'\}\) is the closure (a Kantor set) with measure \(\mu\), \(\mu(\Omega_\Lambda =\mathrm{freq}(P,\Lambda)\)) of the \(O^\Lambda_{\mathrm{trans}} := \{\Lambda -z | z\in \Lambda \}\).
The groupoid \(C^*\)-algebra structure of \(\Lambda\) is introduced the groupoid structure on \(\Omega_{\mathrm{trans}}\) with the convolution product as: two elements \((T-z,T)\) and \((T'-w,T')\) can be composed iff \(T'=T-z\) and in that case \((T-z-w,T-z) * (T-z,T) := (T-z-w,T)\). If \(\theta\) is a 2-cocycle on \(\mathbb R^{2d}\) then on defines a standard 2-cocycle \(\theta_\Lambda((T-z,T),(T'-w,T')) := \theta(z,w).\) For the symplectic cocycle \(\sigma\) on \(\mathbb R^{2d}\), given by \(\sigma((x,\omega),(x'\omega')) := e^{-2\pi i x\omega'}\), one defines the \(C^*\)-algebra \(\mathcal A_\theta = C^*(R_\lambda,\theta)\) of functions on \(R_\Lambda\) by the convolution product \[ f*g(T-z,T) := \sum_{w} f(T-z,T-w)g(T-w,T)\theta((T-z,T-w),(T-w,T)) \] and the involution \(f^*(T-z,T) = \overline{f(T,T-z)}\theta((T-z,T),(T,T-z))\). The \(C^*\)-algebra \(\mathcal A_\theta\) is Morita equivalent to the cross product \(C^*(C(\Omega_\Lambda)\rtimes \mathbb R^{2d}, \theta)\). The condition guarantees that the action of \(\mathbb R^{2d}\) on \(\Omega_\lambda\) is minimal and ergodic, the both \(C^*\)-algebras are simple and have a unique normalized trace given by \[ \mathrm{Tr}(f) = \int_{\Omega_{\mathrm{trans}}} f(T,T)dT, \quad f\in C_c(\Omega_\lambda) \] and following the Birkhoff’s ergodic theorem, it can be written as \[ \mathrm{Tr}(f) = \lim_{k\to \infty} \frac{1}{|\Lambda \cap C_k|}\sum_{z\in\Lambda \cap C_k} f(T-z,T-z). \]
If \(\Lambda \subset \mathbb R^d\) is a “lattice with an aperiodic coloring satisfying the definition of a quasicrystal”, then \(\Omega_\lambda\) is a fiber bundle \(p: \Omega_\Lambda \twoheadrightarrow \mathbb T^d\) over torus \(\mathbb T^d\).
The paper under reviewing is devoted to the problem of description of the image of trace map \(\mathrm{Tr}_* : K_0(\mathcal A_\theta) \to \mathbb R\). The main result of the paper is Theorem 3: The induced map \(p^*: K^0(\mathbb T^d) \rightarrowtail K^0(\Omega_\Lambda)\) is injective and the intersection of the images of \(p^*\) and of \(r_*: K_0(C(\Omega_{\mathrm{trans}}))\to K_0(C(\Omega_{\mathrm{trans}})\rtimes \mathbb Z^d)\cong K_0(C(\Omega_{\Lambda})\rtimes \mathbb R^d)\cong K^0(\Omega_\Lambda)\) induced by \(r: C(\Omega_{\mathrm{trans}}) \to C(\Omega_{\mathrm{trans}}) \rtimes Z^d\), is generated by the class of the trivial bundle. From author’s summary: “In particular, we construct a finitely generated projective module over this algebra, and any multiwindow Gabor frame for \(\Lambda\) can be used to construct an idempotent representing this module in K-theory. For lattice subsets in dimension two, this allows us to prove a twisted version of Bellissard’s gap labeling theorem.”

MSC:

19K56 Index theory
19L50 Twisted \(K\)-theory; differential \(K\)-theory
46L80 \(K\)-theory and operator algebras (including cyclic theory)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Baake, M.; Grimm, U., Aperiodic Order: Volume 1: A Mathematical Invitation, Encyclopedia Math. Appl. (2013), Cambridge University Press · Zbl 1295.37001
[2] Balan, R.; Casazza, P.; Heil, C.; Landau, Z., Density, overcompleteness, and localization of frames. II. Gabor systems, J. Fourier Anal. Appl., 12, 307-344 (2006) · Zbl 1097.42022
[3] Bellissard, J., The noncommutative geometry of aperiodic solids, (Geometric and Topological Methods for Quantum Field Theory (2003), World Sci. Publishing), 86-156 · Zbl 1055.81034
[4] Bellissard, J.; Benedetti, R.; Gambaudo, J.-M., Spaces of tilings, finite telescopic approximation and gap labelings, Comm. Math. Phys., 261, 1-41 (2006) · Zbl 1116.46063
[5] Bellissard, J.; Herrmann, D.; Zarrouati, M., Hulls of aperiodic solids and gap labelling theorems, (Baake, M.; Moody, R., Directions in Mathematical Quasicrystals. Directions in Mathematical Quasicrystals, CRM Monogr. Ser., vol. 13 (2000), Amer. Math. Soc.), 177-206 · Zbl 0972.52014
[6] Benameur, M.; Oyono-Oyono, H., Index theory for quasi-crystals I. Computation of the gap-label group, J. Funct. Anal., 252, 1, 137-170 (2007) · Zbl 1134.46046
[7] Elliott, G., On the \(K\)-theory of the \(C^\ast \)-algebra generated by a projective representation of a torsion-free discrete Abelian group, (Arsene, G.; Strătilă, S.; Verona, A.; Voiculescu, D., Operator Algebras and Group Representations, vol. 1 (1980), Pitman Advanced Publishing Program)
[8] Feichtinger, H. G.; Gröchenig, K., Banach spaces related to integrable group representations and their atomic decompositions, I, J. Funct. Anal., 86, 2, 307-340 (1989) · Zbl 0691.46011
[9] Gahler, F.; Hunton, J.; Kellendonk, J., Integral cohomology of rational projection method patterns, Algebr. Geom. Topol., 13, 1661-1708 (2013) · Zbl 1270.52025
[10] Gillaspy, E., \(K\)-theory and homotopies of 2-cocycles on transformation groups, J. Operator Theory, 73, 465-490 (2015) · Zbl 1399.46103
[11] Gröchenig, K., Foundations of Time-Frequency Analysis (2001), Birkhäuser · Zbl 0966.42020
[12] Gröchenig, K.; Ortega-Cerda, J.; Romero, J. L., Deformations of Gabor systems, Adv. Math., 277, 388-425 (2015) · Zbl 1320.42024
[13] Kaminker, J.; Putnam, I., A proof of the gap labeling conjecture, Michigan Math. J., 51, 537-546 (2003) · Zbl 1054.46047
[14] Kasparov, G. G., Equivariant KK-theory and the Novikov conjecture, Invent. Math., 91, 147-201 (1988) · Zbl 0647.46053
[15] Kellendonk, J., Gap labelling and pressure on the boundary, Comm. Math. Phys., 258, 751-768 (2004) · Zbl 1079.81016
[16] Kellendonk, J., Pattern equivariant functions, deformations and equivalence of tiling spaces, Ergodic Theory Dynam. Systems, 28, 1153-1176 (2008) · Zbl 1149.52017
[17] Lee, J.; Moody, R.; Solomyak, B., Pure point dynamical and diffraction spectra, Ann. Henri Poincaré, 3, 1003-1018 (2002) · Zbl 1025.37004
[18] Luef, F., Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces, J. Funct. Anal., 257, 1921-1946 (2009) · Zbl 1335.46064
[19] Luef, F., Projections in noncommutative tori and Gabor frames, Proc. Amer. Math. Soc., 139, 571-582 (2011) · Zbl 1213.42121
[20] Matei, B.; Meyer, Y., Simple quasicrystals are sets of stable sampling, Complex Var. Elliptic Equ., 55, 947-964 (2010) · Zbl 1207.94043
[21] Meyer, Y., Algebraic Numbers and Harmonic Analysis (1972), North-Holland · Zbl 0267.43001
[22] Muhly, P.; Renault, J.; Williams, D., Equivalence and isomorphism for groupoid \(C^\ast \)-algebras, J. Operator Theory, 17, 3-22 (1987) · Zbl 0645.46040
[23] Renault, J., A Groupoid Approach to \(C^\ast \)-Algebras, Lecture Notes in Math., vol. 793 (1980), Springer · Zbl 0433.46049
[24] Rieffel, M., Projective modules over higher-dimensional noncommutative tori, Canad. J. Math., 40, 257-338 (1988) · Zbl 0663.46073
[25] Sadun, L., Topology of Tiling Spaces, Univ. Lecture Ser., vol. 46 (2008), American Mathematical Society · Zbl 1166.52001
[26] Sadun, L.; Williams, R. F., Tiling spaces are Cantor set fiber bundles, Ergodic Theory Dynam. Systems, 23, 307-316 (2003) · Zbl 1038.37014
[27] Shechtman, D.; Blech, I.; Gratias, D.; Cahn, J., Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett., 53, 1951-1953 (1984)
[28] van Elst, A., Gap labelling for Schrödinger operators on the square and cubic lattices, Rev. Math. Phys., 6, Article 319 pp. (1994) · Zbl 0804.47053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.