Gabor Frames for Quasicrystals II: Gap Labeling, Morita Equivale

Gabor Frames for Quasicrystals II: Gap Labeling, Morita Equivalence, and Dual Frames University of Maryland June 11, 2015

Overview Twisted Gap Labeling

Outline Twisted Gap Labeling

Physical Quasicrystals In 1984, Shechtman et. al. discovered materials whose atoms did not have the structure of a lattice, but instead had the structure of a quasicrystal. How can we study the electron interactions in such a material? We consider a Schrödinger operator of the form H Λ = 1 ( ) 2 2m i ea + V acting on L 2 (R d ), where V is multiplication by a pattern equivariant function. The vector potential A models the effect of a constant, uniform magnetic field.

Physical Quasicrystals (cont) In general, the Schrödinger operator H Λ will have gaps in its spectrum, and we would like to know the energy levels at these gaps. The location of the gaps depends on the specific potential, but the possible values of the energy levels are constrained by the structure of Λ. We can consider the twisted groupoid C -algebra A θ where θ is a cocycle on R d corresponding to the choice of magnetic field. When θ = σ, the symplectic cocycle, we get the algebra A we defined yesterday. There is a trace defined on A θ, and we define the gap labeling group to be Tr (K 0 (A θ )) R. As long as the potential is modeled on Λ, the energy levels of the Schrödinger operator can only take values in the gap labeling group.

Gap Labeling Conjecture (Bellissard s Gap Labeling Conjecture) When the magnetic field θ = 0, the gap labeling group is generated by the frequencies of patches in Λ.

Gap Labeling Conjecture (Bellissard s Gap Labeling Conjecture) When the magnetic field θ = 0, the gap labeling group is generated by the frequencies of patches in Λ. There are now multiple proofs of this conjecture. All use some version of index theory.

Main Question Question Can we determine the gap labeling group when a magnetic field is present? Strategy: Find a systematic way to construct modules over A θ, then show that these modules generate all of K 0 (A θ ).

Computing K-theory K 0 (A θ ) = K 0 (A θ=0 ), and it is easy to identify elements which come from projections onto clopen sets in the transversal. Thus the patch frequencies will still be contained in the gap labeling group for A θ. The difficulty is in identifying the other elements of K 0 (A θ ) and computing their traces. (Recall that when θ = 0, all these elements end up having trace equal to 0.)

Twisted Gap Labeling for Marked Lattices As a first attempt, let s assume Λ is a lattice with an aperiodic coloring. Then the groupoid C -algebra of Λ can be written as a crossed product by an action of Z d on the transversal. This yields a map i from C r (Z d ) = C(T d ) into A θ=0. When we twist by a cocycle θ on R d, we get instead a map from the noncommutative torus A θ into A θ. Theorem The map i : A θ A θ preserves the trace and is injective on K 0. Aside from the trivial class, the image of this map on K 0 does not overlap with classes coming from clopen sets in the transversal.

Twisted Gap Labeling for Marked Lattices (cont) Corollary When Λ is a marked lattice, the gap labeling group for A θ contains the image of the trace map on the noncommutative torus A θ.

Twisted Gap Labeling for Marked Lattices (cont) Corollary When Λ is a marked lattice, the gap labeling group for A θ contains the image of the trace map on the noncommutative torus A θ. Any quasicrystal can be deformed to a marked lattice in such a way that the corresponding transversal groupoids are isomorphic. Thus the tiling space is a fiber bundle over the torus with Cantor fiber p : Ω Λ T d. Corollary The induced map p is injective on K 0, and its image does not overlap with projections onto clopen sets in the transversal, except for the trivial class.

Interpretation in Terms of H Λ We can understand the image of the map i using the modules we constructed yesterday. For the symplectic cocycle σ, we constructed a module H Λ over A σ. Applying a linear map T to Λ does not change the transversal groupoid, but it does change the way that the cocycle σ sits inside H 2 (R trans ). Thus H T Λ is a module over a A θ where θ = T σ. Using this technique, we can construct modules over A θ for any θ. We can also construct many modules over A θ for any fixed θ by finding maps such that T σ = θ. The image of i consists of all such modules.

Constructing All Modules Unfortunately, the classes in the image of the map i, along with those coming from clopen sets in the transversal, do not in general exhaust K 0 (A θ ). What if we expand our scope, applying not just linear maps, but more general deformations which preserve the transversal groupoid? Question Can all classes in K 0 (A θ ) be represented as H DΛ where D is a deformation in the sense of Clark/Sadun/Kellendonk? If so, we could compute the twisted gap labeling group from knowledge of how the deformations change the density of Λ.

Twisted Gap Labeling in Dimension Two In dimension two we can begin to get more complete results, because in dimension two K 0 actually is generated by the image of i, along with projections from the transversal. In this case the cocycle θ is determined by a single nonzero parameter (also denoted θ) and K 0 (A θ ) = Z +θ Z. Thus we have: Theorem Suppose Λ is a marked lattice in R 2. Then the twisted gap labeling group of A θ is generated by the patch frequencies of Λ, along with a single other number θvol(λ).

Twisted Gap Labeling for 2-D Lattice Subsets Corollary Let Λ be a quasicrystal which is a subset of a lattice in R 2. Then the twisted gap labeling group is generated by the patch frequencies, along with a single other number θ Dens(Λ). Although any quasicrystal can be deformed to a lattice subset, it is unclear how such a deformation transforms a cocycle θ. Theorem (Julien/K.) If a deformation D is given by an asymptotically negligible class in H 1 (R trans ) then D σ = σ.

Twisted Gap Labeling for Common Examples Corollary Suppose Λ R 2 is a cut and project set or a Pisot substitution. Then the twisted gap labeling group is generated by the patch frequencies of Λ and Conjecture θ Dens(Λ). Suppose Λ is a two dimensional quasicrystal. Then the twisted gap labeling group is generated by the patch frequencies of Λ and θ Dens(Λ). Question What about higher dimensions, or when the cocycle θ is not the restriction of a cocycle on R 2d?

Outline Twisted Gap Labeling

NC Tori Recap Recall that a noncommutative torus A L is generated by time frequency shift operators taken from a lattice L acting on M 1 (R d ). When G(g, L) is a Gabor frame for L, the frame operator S L commutes with the action of A L, so is contained in the commutant of the action. The commutant is also a rotation algebra A L, where L is the adjoint lattice. Thus S L has an expansion in terms of time-frequency shifts from L, which we know as the Janssen representation.

Quasicrystal Algebras Recap Recall that we constructed a representation of the C -algebra A 1 on C(Ω trans, M 1 (R d )). We can think of A 1 as being generated by partial time-frequency shifts of the form I(χ z )Ψ(T ) = { π(z)ψ(t z) if z T 0 otherwise If G(g, Λ) is a Gabor frame, how does the frame operator fit into this picture?

Global Frame Operator We can define the global frame operator S as an operator on C(Ω trans, M 1 (R d )) by (SΨ)(T ) = S T Ψ(T ). The global frame operator S commutes with all operators in A.

Global Frame Operator We can define the global frame operator S as an operator on C(Ω trans, M 1 (R d )) by (SΨ)(T ) = S T Ψ(T ). The global frame operator S commutes with all operators in A. Similar to the case of lattice Gabor frames, we can try to find generators for End A H Λ, then express S in terms of those generators.

Eigenvalues and Eigenfunctions A function e k C(Ω Λ ) is an eigenfunction with eigenvalue k if e k (T z) = e 2πik z e k (T ) for all z R 2d. When an eigenfunction exists for a given eigenvalue, it is unique. We define the Bohr coefficient c k by 1 c k := lim e 2πik z. R vol(b R ) z Λ B R We have c k = 0 when k is not an eigenvalue. The c k are averages of exponential functions over the quasicrystal, and are also equal to the integrals of eigenfunctions over the transversal.

Eigenfunction Operators We can define operators τ k on C(Ω trans, M 1 (R d )) by (τ k Ψ)(T ) = e k (T )π(ǩ)ψ(t ) where k R 2d, e k is an eigenfunction with eigenvalue k, and ǩ is a symplectic transformation of the eigenvalue k : ( ) 0 I ǩ = k I 0 The operators τ k commute with all operators in A. We can think of them as fiberwise time-frequency shifts. They generate either a noncommutative torus or noncommutative solenoid inside End A H Λ.

Janssen Representation Although the operators τ k will never generate all of End A H Λ, we can still try to find an expansion of S in terms of the τ k using the trace on End A H Λ. We can compute Tr(Sτ k ) = c k N g i, π(ǩ)g i i=1 where g i are the windows of the frame. This suggests S = k Eig(Λ) i=1 N c k g i, π(ǩ)g i τ k. Unfortunately, this sum will not converge absolutely. Properly interpreted, this identity is true for cut and project sets.

Non-existence of Tight Frames Even though the previous expansion of the global frame operator does not always hold, we can use the trace computation to prove a non-existence result for tight multiwindow frames. Theorem Let Λ R 2d be a quasicrystal. Assume that the group of continuous eigenvalues Eig(Λ) is dense in R 2d, and that the Fourier-Bohr coefficients c k are non-zero in a neighborhood of 0. Also assume that the operators τ k generate a simple noncommutative torus or solenoid. Then it is not possible to find a tight multiwindow frame G(g 1,..., g N, Λ) where all g i M 1 (R d ).

Removing Some Assumptions I am inclined to believe that if the collection of eigenvalues is dense then the Fourier-Bohr coefficients must be non-zero in a neighborhood of the identity (this is true for cut and project sets). Perhaps this assumption always holds. We could remove the simplicity assumption on the algebra generated by the τ k if we could compute lim R 1 Λ B R π(ǩ)(sλ g ) 1 2 π(z)g, (S Λ g ) 1? 1 2 π(z)g = Dens(Λ) δ k,0 z Λ B R whenever g M 1 (R d ) generates a Gabor frame for Λ and k is an eigenvalue.

Outline Twisted Gap Labeling

Lattice Dual Frames When L is a lattice, the frame operator S L commutes with time frequency shifts from L. Dual frame elements have the form g z = π(z)s 1 L g. What is the structure of the dual frame for a quasicrystal?

Using the Covariance Condition Suppose G(g, Λ) is a Gabor frame. In general, we only have S Λ π(z) = π(z)s Λ z. We can rewrite the dual frame elements as g z = S 1 Λ π(z)g = π(z)s 1 Λ z g.

Using the Covariance Condition Suppose G(g, Λ) is a Gabor frame. In general, we only have S Λ π(z) = π(z)s Λ z. We can rewrite the dual frame elements as g z = S 1 Λ π(z)g = π(z)s 1 Λ z g. But the functions S 1 Λ zg are exactly the functions you get by applying S 1 to the constant section Ψ g!

Approximating with Multi-Frames Since S 1 Ψ g is a continuous function on the transversal (a Cantor set) with values in M 1 (R d ), it can be approximated by a locally constant function. This amounts to saying that the dual frame elements can be approximated by a Gabor multiframe! 1. Cut transversal into n pieces. 2. For each piece choose a translate of Λ in that piece, say Λ z 1,... Λ z n and estimate each of S 1 Λ z i g. 3. To estimate S 1 Λ zg for other z, just check which piece Λ z lies in and use the function we already computed. 4. Now we can estimate the dual frame elements π(z)s 1 Λ z g for all z Λ!

Abstract Characterization Definition Let G = {g z } z Λ M 1 be a frame sequence where Λ is a quasicrystal. Then we call G a pattern equivariant Gabor system if the following condition holds: for every R > 0 there exists ɛ R > 0 so that whenever we have two points z, z Λ such that Λ z and Λ z agree on a ball of radius R then and ɛ R 0 as R. π(z) 1 g z π(z ) 1 g z M 1 < ɛ R

Open Questions Can we realize End A (H Λ ) as a twisted quasicrystal algebra? If Λ has pure discrete spectrum, is the von Neumann closure of End A (H Λ ) generated by the eigenfunction operators τ k? These questions should be related to understanding the Fourier transform of the point measure λ Λ δ λ. Given a class c in H 2 (R trans ), can we always find a deformation D such that D σ = c? My guess is no, but finding a counterexample would be instructive. Can we describe all deformations which fix the symplectic cocycle (i.e. D σ = σ)? Can we find an example where using more general deformations allows us to construct modules which do not come from noncommutative tori?

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