Gabor frames and Wannier bases from groupoid Morita equivalences
aa r X i v : . [ m a t h . OA ] S e p GABOR FRAMES AND WANNIER BASES FROM GROUPOID MORITAEQUIVALENCES
C. BOURNE AND B. MESLAND
Abstract.
Given a groupoid equivalence between Hausdorff groupoids where the right groupoidis ´etale, we construct Gabor frames of the Hilbert space localisations of the Morita equivalencebimodule of the reduced groupoid C ∗ -algebras. For finitely generated and projective submod-ules, we show these Gabor frames are orthonormal bases if and only if the module is free.We then apply this result to the study of localised Wannier bases of spectral subspaces ofSchr¨odinger operators with atomic potentials supported on (aperiodic) Delone sets. The non-commutative Chern numbers provide a topological obstruction to fast-decaying Wannier basesand we show this result is stable under deformations of the underlying Delone set. Introduction
A key question in Gabor analysis is the reconstruction of elements in a Hilbert space h via aGabor frame, a set of vectors { T j w , . . . , T j w m } j ∈ J spanning h , where { T j } j ∈ J ⊂ B ( h ) is somecanonically defined set of operators (typically translation and modulation). Morita equivalencebimodules of C ∗ -algebras, also called imprimitivity bimodules, have been shown to be a usefultool in the construction of Gabor frames coming from locally compact abelian groups. See [22,1, 2] for further details.A similar concept to Gabor frames arising from physics are Wannier bases. Given a Schr¨odinger-type operator H with periodic potential acting on L ( R d , C n ), a Wannier basis is an orthonormalbasis of a spectral subspace of H constructed from a finite set of functions along with theirtranslations in Z d . Because the operator H has a periodic potential, such bases exist by theBloch–Floquet transform. The regularity of Wannier bases change drastically depending onthe topological properties of the spectral band of the Schr¨odinger operator, where a delocalisedWannier basis can be used as an indicator that the system has a non-trival topological phase,see [12, 24, 26] for example.In the context of periodic systems with a space group translation symmetry G ⊂ R d , it wasshown by Ludewig and Thiang that the existence of a fast-decaying Wannier basis is equivalentto whether a finitely generated and projective C ∗ r ( G )-module is free or not [21].The purpose of this paper is twofold. First, using a similar framework to [2], we study therelation between Gabor frames and Morita equivalence bimodules arising from groupoid equiv-alences. We then use this relation to extend the results of Ludewig and Thiang [21] on fast-decaying Wannier bases to ´etale groupoids. Regularity of frames is examined using pre-Moritaequivalence bimodules of algebras defined from derivations and differential seminorms.For both of our aims, our guiding example is the ´etale groupoid G Del constructed from a Deloneset Λ ⊂ R d and which is equivalent to the crossed product groupoid of the translation actionon the orbit space of Λ, Ω Λ ⋊ R d . In previous work [9, 10], the index theory of G Del wasstudied and its application to aperiodic topological phases. In [20], Gabor frames of L ( R d )were constructed from Delone subsets of R d with finite local complexity and the groupoid G Del using results from [16]. We remark that we do not consider the important question of Gaborduality in this context.
Outline and main results.
Because our results bring together constructions from Gaboranalysis, groupoids, C ∗ -modules and Morita equivalence, we give a brief overview of these Date : September 30, 2020. oncepts in Section 2. In Section 2.3, using the framework of differential seminorms (cf. [8]),we construct pre-Morita equivalence bimodules for pre- C ∗ -subalgebras obtained from a finitefamily of commuting unbounded ∗ -derivations.In Section 3 we consider a groupoid equivalence G ← Z → H of Hausdorff, second countableand locally compact groupoids G and H , where H is ´etale with a compact unit space H (0) .By choosing the evaluation state for some x ∈ H (0) we obtain a translation action of the fibre r − ( x ) on the Hilbert space localisation h x of the Morita equivalence bimodule between C ∗ r ( G )and C ∗ r ( H ). This allows us to construct a normalised tight frame of h x from the C ∗ -moduleframe of the Morita equivalence bimodule. Restricting to finitely generated and projective C ∗ r ( H )-submodules, we obtain a multi-window Gabor frame for a subspace of the Hilbert spacelocalisation h x . This frame is an orthonormal basis for all x ∈ H (0) if and only if the finitelygenerated and projective module is free, and thus its class in the reduced K -theory of C ∗ r ( H )is trivial. These results are extended in Section 4 to the case of abstract transversals with anormalised 2-cocycle twist.We apply these results to the ´etale Delone groupoid in Section 5. We consider a magneticSchr¨odinger operator with an atomic potential v arranged on a Delone set Λ ⊂ R d , H Λ = d X j =1 (cid:16) − i ∂∂x j − A j (cid:17) + X p ∈ Λ v ( · − p ) , where A is the magnetic potential. Results by Bellissard et al. show that for sufficiently regular v , H Λ and its magnetic translates are affiliated to the crossed product C ∗ -algebra C ∗ r (Ω Λ ⋊R d , θ ) with θ a magnetic twist [4, 5]. We show that for any Delone set L in the transversalsubset Ω ⊂ Ω Λ , a gapped spectral subspace of H L has a normalised tight frame built fromthe magnetic translations { U y } y ∈L . This frame is an orthonormal basis if and only if thecorresponding finitely generated and projective C ∗ r ( G Del , θ )-module is free. Using derivationsof the groupoid algebras and differential seminorms, we show this normalised tight frame hasfaster than polynomial decay. We therefore see that topological properties of spectral subspacesof the Delone Schr¨odinger operator can be related to the reguarity of (aperiodic) Wannier bases.The regularity of such Wannier basis is closely related to the Localisation Dichotomy Conjecturefor non-periodic insulators raised in [24, Section 5 (arXiv version)] and further studied in [25].We prove a weaker version of this conjecture in Section 5.3 and show that the existence ofWannier bases with faster than polynomial decay is equivalent to the existence of Wannierbases such that P j (1 + | x | ) | w j ( x ) | ∈ L ( R d ). This in turn is equivalent to the spectralprojection defining a freely generated C ∗ r ( G Del , θ )-module. Our results are weaker than thoseposed in [24] as we consider a family of Schr¨odinger operators and Hilbert spaces rather than asingle Hamiltonian. Similarly we do not consider Wannier bases with exponential decay.By relating the existence of a localised Wannier basis to a K -theoretic property, the noncommu-tative Chern numbers for C ∗ r (Ω Λ ⋊ R d , θ ) and C ∗ r ( G Del , θ ) studied in [9, 10, 11] give a topologicalobstruction to a Wannier basis with fast decay. We also show that these Chern number formulasare continuous under deformations of the magnetic field or Delone atomic potential providedthe spectral gap stays open. This implies that a non-localised Wannier basis is stable underdeformations of the atomic potential (e.g. from a periodic lattice to an aperiodic point pattern).2.
Preliminaries
Gabor frames and analysis.
Let us recall a few basic definitions from Gabor analysis.
Definition 2.1.
Let h be a Hilbert space and { g j } j ∈ J is a collection of elements in h . We saythat { g j } j ∈ J form a Hilbert space frame if there are constants C, D > C k ψ k ≤ X j ∈ J (cid:12)(cid:12) h g j , ψ i (cid:12)(cid:12) ≤ D k ψ k , ψ ∈ h . f C = D , then { g j } j ∈ J is called a tight frame. If C = D = 1, then { g j } j ∈ J is called a normalisedtight frame or Parseval frame.Orthonormal bases are obvious examples of normalised tight frames. The restriction of anorthonormal basis of a Hilbert space to a closed subspace yields a normalised tight frame forthe subspace. Normalised tight frames always arise as a compression of an orthonormal basis. Proposition 2.2 ([17], Section 1) . Let { g j } j ∈ J be a normalised tight frame of a Hilbert space h . Then there is Hilbert space h and an orthonormal basis { e j } j ∈ J of h ⊕ h such that g j = pr ( e j ) . Any Hilbert space frame yields an invertible synthesis map S : h → h , S ( ψ ) = X j ∈ J g j h g j , ψ i . We note that, in contrast to the Gabor analysis literature, we work with inner-products that arelinear on the right. This is to make our results more easily compatible with right C ∗ -modulesand their Hilbert space localisations (Definition 2.11 below). Because S is invertible, one obtainsa reconstruction formula for elements in h . ψ = X j ∈ J S − g j h g j , ψ i , ψ ∈ h . The elements { S − g j } j ∈ J are called the dual frame to { g j } j ∈ J .2.2. Pre- C ∗ -modules and Morita equivalence. Following a similar framework to [2], we willuse C ∗ -modules and Morita equivalence bimodules to study questions around Gabor frames oftheir Hilbert space localisation. We now recall some basic definitions and establish notation.Further details can be found in [7, 29].We will also be interested in the case where the C ∗ -algebras A and B have dense ∗ -subalgebras A and B . Definition 2.3.
Let B be a C ∗ -algebra, B ⊂ B a dense ∗ -subalgebra and E B a vector spacethat is also a right B -module. We say that E B is a right inner product B -module if there is abilinear pairing ( e , e ) ( e | e ) B ∈ B such that for e , e , e ∈ E ( e | e ) B = ( e | e ) ∗B , ( e | e · b ) B = ( e | e ) B b, ( e | e ) B ≥ ∈ B, ( e | e ) B = 0 ⇒ e = 0 . An inner product module E B is called full if span { ( e | e ) : e , e ∈ E B } is C ∗ -norm dense in B . If B is a C ∗ -algebra, a C ∗ -module is a right inner product B -module that is complete in thenorm k e k := k ( e | e ) B k B .On an inner product module E B , the norm k e k := k ( e | e ) B k B is defined and the completionof E B in this norm is a right C ∗ -module E B over B . For a right inner product B -module, the ∗ -algebra of finite rank operators Fin B ( E B ) is defined to be the algebraic span of the finite-rankoperators { Θ Re ,e } e ,e ∈E , whereΘ Re ,e ( e ) = e · ( e | e ) B , (cid:0) Θ Re ,e (cid:1) ∗ := Θ Re ,e , e , e , e ∈ E . When E B is a C ∗ -module over a ∗ -algebra B , the compact endomorphisms K B ( E ) are defined asthe operator norm closure of Fin B ( E ). It is a closed two-sided ideal in the C ∗ -algebra End ∗ B ( E )of adjointable operators on E B . Definition 2.4.
Let E B be a right inner product B -module. A set { e , · · · , e n } is called a finitemodule frame if Id E = n X k =1 Θ Re k ,e k . f E B is a right C ∗ -module, a countable subset { e j } j ∈ J ⊂ E B is a (right) C ∗ -module frame if P j Θ Re j ,e j converges strictly to Id E .We remark that any countably generated right C ∗ -module over a σ -unital algebra B admits a C ∗ -module frame [7, 29]. If an inner product module E B admits a finite frame { v j } nj =1 , thenthere is a projection p = p ∗ = p ∈ M n ( B ) and right module maps S : E → B n , R : B n → E , (1) S ( e ) = (cid:0) ( v j | e ) B (cid:1) nj =1 , R (cid:0) ( b j ) nj =1 (cid:1) = n X j =1 v j · b j , (2)that restrict to isomorphisms S : E → p B n and R : p B n → E . In particular p = ( v i | v j ) B ∈ M n ( B ) , R ◦ S = Id E , e = R ◦ S ( e ) = n X j =1 v j · ( v j | e ) B = n X j =1 Θ Rv j ,v j ( e ) . Similar formulas hold for left modules with a finite frame.
Definition 2.5.
Let A and B be C ∗ -algebras. An A - B Morita equivalence bimodule is a fullright-
B C ∗ -module and full left- A C ∗ -module A E B such that( a · e | e ) B = ( e | a ∗ · e ) B , A ( e | e · b ) = A ( e · b ∗ | e ) , A ( e | e ) · e = e · ( e | e ) B for all a ∈ A , b ∈ B and e , e , e ∈ E . We say that A and B are Morita equivalent if there is aMorita equivalence bimodule A E B .To distinguish left and right inner products, for e , e ∈ E we use the notationΘ Le ,e ( e ) = A ( e | e ) · e , Θ Re ,e ( e ) = e · ( e | e ) B . A full right-
B C ∗ -module is a Morita equivalence bimodule between K B ( E ) and B with the K B ( E )-valued inner product K ( E ) ( e | e ) = Θ Re ,e . Hence A is Morita equivalent to B if andonly if there is a full right- B C ∗ -module E B and a ∗ -isomorphism φ : A → K B ( E ). Definition 2.6.
Let A and B be dense ∗ -subalgebras of C ∗ -algebras A and B . A pre-Moritaequivalence bimodule is an A - B bimodule E , equipped with a full left- A valued and a full right- B valued inner product such that for any a ∈ A , b ∈ B and e, e , e , e ∈ E the compatibilityconditions A ( e · b | e · b ) ≤ k b k A ( e | e ) , ( a · e | a · e ) B ≤ k a k ( e | e ) B , A ( e | e ) · e = e · ( e | e ) B , are satisfied. Here k · k denotes the C ∗ -norm on the algebras A and B , and the inequalities arein the C ∗ -algebras A and B .As expected, a pre-Morita equivalence bimodule A E B can be completed to a Morita equivalencebimodule A E B , using either of the norms k e k := k ( e | e ) B k B or k e k := k A ( e | e ) k A see [29,Section 3.1]. We proceed with some definitions and results concerning frames in dense submod-ules of C ∗ -modules. For this we need our dense ∗ -subalgebras to be equipped with additionalanalytic structure. Definition 2.7.
We say that a ∗ -algebra A is a pre- C ∗ -algebra if it is(i) Fr´echet, i.e. complete and metrizable such that the multiplication is jointly continuous;(ii) Isomorphic to a proper dense ∗ -subalgebra ι ( A ) of a C ∗ -algebra A , where ι : A ֒ → A isthe inclusion map, and ι ( A ) is stable under the holomorphic functional calculus. Thatis, if f is a holomorphic function on a neighbourhood of the spectrum of a ∈ ι ( A ), then f ( a ) ∈ ι ( A ).Stability under the holomorphic functional calculus extends to nonunital algebras, since thespectrum of an element in a nonunital algebra is defined to be the spectrum of this element inthe one-point unitization, though we must restrict to functions satisfying f (0) = 0. Similarly,the definition of a Fr´echet algebra does not require a unit. roposition 2.8 ([31]) . If A is a pre- C ∗ -algebra with C ∗ -completion A , then the map inducedby the inclusion ι ∗ : K j ( A ) → K j ( A ) is an isomorphism. Lemma 2.9.
Let A and B be pre- C ∗ -algebras with B unital and A E B a pre-Morita equivalencebimodule. (1) There is a finite left module frame { g , . . . , g n } ⊂ E and B = P nk =1 ( g k | g k ) B . (2) For any p = p ∗ = p ∈ M n ( A ) , p E ⊕ n is a finitely generated and projective B -module andthere exists a finite right module frame { v , · · · , v m } ⊂ p E ⊕ n .Proof. Write A and B for the C ∗ -algebra closures of A and B and E for the C ∗ -closure of E ,which is a Morita equivalence bimodule for the C ∗ -algebras A and B .(1) The ∗ -algebra of finite rank operators span { Θ Le ,e } e ,e ∈E is C ∗ -norm dense in A K ( E ). Thusfor ε < { e , · · · , e n } ⊂ E such that the operator g := n X k =1 Θ Le k ,e k = n X k =1 ( e k | e k ) B ∈ B , satisfies k B − g k B < ε . Hence the positive element g is invertible in B , and thus it is invertiblein B . Since B is stable under holomorphic functonal calculus we have g − ∈ B . Define g k := e k · g − ∈ E , for 1 , · · · , n so that n X k =1 ( g k | g k ) B = n X k =1 Θ Lg k ,g k = n X k =1 Θ Le k g − ,e k g − = g − n X k =1 Θ Le k ,e k ! g − = g − gg − = 1 B , which proves the claim.(2) Since p is a compact operator on E ⊕ n , it is finite rank by [15, Corollary 3.10], so themodule W := pE ⊕ n is finitely generated and projective over B . Write W := p E ⊕ n for the dense B -submodule defined by p ∈ M n ( A ).The ∗ -algebra of finite rank operators span { Θ Re ,e } e ,e ∈W is C ∗ -norm dense in K B ( W ). Hencefor ε < w , · · · , w m ∈ W such that w := P Θ Rw k ,w k satisfies k p − w k K B ( W ) < ε .It follows that w is invertible in the unital C ∗ -algebra K B ( W ) ∼ = pM n ( A ) p and has spectrumcontained in B (1; ε ). The spectrum of w in M n ( A ) is thus contained in the disconnected set B (1; ε ) ∪ { } . By spectral invariance, the same holds for the spectrum of w ∈ M n ( A ). Thusthere is a function f , holomorphic on a neighborhood of the spectrum of w such that f (0) = 0and f ( z ) = z − for z ∈ B (1; ε ). Hence f ( w ) ∈ M n ( A ) ∩ pM n ( A ) p = pM n ( A ) p and satisfies f ( w ) wf ( w ) = p . Now put v k := f ( w ) w k so that, as above, m X k =1 Θ Rv k ,v k = f ( w ) m X k =1 Θ Rw k ,w k ! f ( w ) = p, which proves the claim. (cid:3) Derivations, pre-Morita equivalence bimodules and localisation.
We now describea general method to construct pre-Morita equivalence bimodules and pre- C ∗ -algebras from afamily of densely defined derivations on a given C ∗ -algebra.As a technical tool we will use the notion of differential seminorms introduced in [8, Definition3.1]. The space ℓ ( N ) is an algebra in the convolution product f ∗ g ( n ) := P k ≤ n f ( k ) g ( n − k ) . Thesubspace ℓ ( N ) := ℓ ( N , R + ) ⊂ ℓ ( N ) inherits the coordinatewise ordering from R + and satisfies ℓ ( N ) ℓ ( N ) ⊂ ℓ ( N ). Following [8], a differential seminorm on a subalgebra A ⊂ A is a map T : A → ℓ ( N ) such that T ( a )(0) ≤ C k a k and T ( λa ) = | λ | T ( a ) and T ( ab ) ≤ T ( a ) ∗ T ( b ). Thefunctional R : ℓ ( N ) → C , f P k ∈ N f ( k ), is positive and multiplicative, and if T : A → ℓ ( N )is a differential seminorm, then R T : A → R + is a submulitiplicative seminorm. roposition 2.10 (Cf. [8]) . Let A be a C ∗ -algebra with norm k · k , A ⊂ A a dense ∗ -subalgebraand { ∂ j : A → A } dj =1 a finite family of commuting ∗ -derivations. Then for α ∈ N d a multi-index, k a k n = X | α |≤ n (cid:13)(cid:13) ∂ α a (cid:13)(cid:13) α ! , ∂ α = ∂ α · · · ∂ α d d , | α | := n X k =1 α k , α ! := d Y k =1 ( α k !) , is a sequence of submutliplicative seminorms on A . Let A n denote the closure of A in theseminorms k · k k , k ≤ n and A ∞ := lim ←− A n the Fr´echet closure of A in all the seminorms k · k n .Then for n = 0 , · · · , ∞ , A n is a pre- C ∗ -algebra.Proof. Defining (cid:18) αβ (cid:19) := Q dk =1 (cid:18) α k β k (cid:19) , one proves that ∂ α ( ab ) = P β ≤ α (cid:18) αβ (cid:19) ∂ β ( a ) ∂ α − β ( b ) , byinduction on α . Following [8], for n ∈ N we consider the maps T n : A → ℓ ( N ) , T n ( a )( k ) := P | α | = k k ∂ α a k α ! for k ≤ n, T n ( a )( k ) = 0 for k > n. Indeed the T n satisfy T n ( a )(0) = k a k and T n ( λa ) = | λ | T n ( a ) as well as T n ( ab )( k ) = X | α | = k α ! (cid:13)(cid:13) ∂ α ( ab ) (cid:13)(cid:13) ≤ X | α | = k X β ≤ α β ! (cid:13)(cid:13) ∂ β ( a ) (cid:13)(cid:13) α − β )! (cid:13)(cid:13) ∂ α − β ( b ) (cid:13)(cid:13) ≤ X | α | + | β | = k β ! (cid:13)(cid:13) ∂ β ( a ) (cid:13)(cid:13) α ! (cid:13)(cid:13) ∂ α ( b ) (cid:13)(cid:13) ≤ T n ( a ) ∗ T n ( b )( k ) , which shows that T n is a differential algebra norm. Moreover for k ≥ T n ( ab )( k ) ≤ X | α | + | β | = k β ! (cid:13)(cid:13) ∂ β ( a ) (cid:13)(cid:13) α ! (cid:13)(cid:13) ∂ α ( b ) (cid:13)(cid:13) = k a k (cid:16) X | β | = k β ! (cid:13)(cid:13) ∂ β ( b ) (cid:13)(cid:13)(cid:17) + (cid:16) X | α | = k α ! (cid:13)(cid:13) ∂ α ( a ) (cid:13)(cid:13)(cid:17) k b k + X | α | , | β |≥ β ! (cid:13)(cid:13) ∂ β ( a ) (cid:13)(cid:13) α ! (cid:13)(cid:13) ∂ α ( b ) (cid:13)(cid:13) , so that each T n is of logarthmic order ≤ k a k n = R T n ( a )and the result for A n follows from [8, Propositions 3.3 and 3.12]. The result for A ∞ then followssince A ∞ = T ∞ n =0 A n and the result holds for each A n . (cid:3) Now let A ⊂ A and B ⊂ B be dense ∗ -subalgebras and E an A - B pre-Morita equivalencebimodule. Suppose we are given a families of commuting ∗ -derivations { ∂ Lj } dj =1 : A → A , { ∂ Rj } dj =1 : B → B , as well as a commuting family of operators ∇ j : E → E such that for each j and all a ∈ A , ξ, η ∈ E and b ∈ B we have ∇ j ( a · ξ · b ) = ∂ Lj ( a ) · ξ · b + a · ∇ j ( ξ ) · b + a · ξ · ∂ Rj ( b ) , (3) ∂ Rj ( ξ | η ) B = ( ξ | ∇ j ( η )) B − ( ∇ j ( ξ ) | η ) B . (4)It is worth noting that the identity ∂ Lj A ( ξ | η ) = A ( ξ | ∇ j ( η )) − A ( ∇ j ( ξ ) | η ) , (5)is satisfied as well. By using Equations (3), (4) and the compatibility of left and right innerproducts, we find for ξ, η, e ∈ E : A ( ξ | ∇ j ( η )) · e − A ( ∇ j ( ξ ) | η ) · e = ξ · ( ∇ j ( η ) | e ) B − ∇ j ( ξ ) · ( η | e ) B = ξ · ( η | ∇ j ( e )) B − ξ · ∂ Rj ( η | e ) − ∇ j ( ξ ) · ( η | e ) B = ξ · ( η | ∇ j ( e )) B − ∇ j ( ξ · ( η | e ) B )= A ( ξ | η ) · ∇ j ( e ) − ∇ j ( A ( ξ | η ) · e ))= ∂ Lj A ( ξ | η ) · e, o that (5) follows.We write A n , B n for the pre- C ∗ -algebra completions obtained through Proposition 2.10. Simi-larly we write ∇ α := ∇ α · · · ∇ α d d , α ∈ N d , k e k n := X | α |≤ n k∇ α ( e ) k α !as well as E n for the completion of E in the seminorms up to degree n , and E = E ∞ for thecompletion of E in all these seminorms. Definition 2.11.
Suppose that E B is a right C ∗ -module with B a unital C ∗ -algebra and ω B : B → C a (unital) state. The localisation h ω is the Hilbert space that comes from thecompletion of E in the inner product h e , e i ω = ω B (cid:0) ( e | e ) B (cid:1) . Remark . If A E B is a Morita equivalence bimodule and the state τ : B → C is a trace, thenthere is a canonical dual tracial weight Tr τ on K B ( E ) ∼ = A such that Tr τ (Θ Re ,e ) = τ (( e | e ) B )for any e , e ∈ E . We can localise A E B with a left-linear inner-product from Tr τ to obtainthe dual localisation space h ∗ ω . Hence in this special case, the localisation Hilbert space directlyagrees with the Gabor analysis literature. See [2] for more details.Given any state ω : B → C , the seminorms k · k n induce a family of seminorms k · k n,ω on theHilbert space localisation h ω , k ξ k n,ω := X | α |≤ n k∇ α ( ξ ) k h ω α ! , n ∈ N . Proposition 2.13.
For n = 0 , · · · , ∞ the space E n is a A n - B n pre-Morita equivalence bimodule.Moreover, for any state ω : B → C , the image of E n in h ω is finite in the seminorms k · k k,ω for ≤ k ≤ n .Proof. The space L ( E ) := (cid:26)(cid:18) a ξη ∗ b (cid:19) : a ∈ A , b ∈ B , ξ, η ∈ E (cid:27) with multiplication and involution (cid:18) a ξ η ∗ b (cid:19) · (cid:18) a ξ η ∗ b (cid:19) := (cid:18) a a + A ( ξ | η ) a ξ + ξ b ( a ∗ η ) ∗ + ( η b ) ∗ ( η | ξ ) B + b b (cid:19) , (cid:18) a ξη ∗ b (cid:19) ∗ := (cid:18) a ∗ η ∗ ξ b ∗ (cid:19) , is an associative ∗ -algebra, the linking algebra of E . It is a dense ∗ -subalgebra of the linking C ∗ -algebra L ( E ) of the C ∗ -module closure E of E . Using the identities (3), (4) and (5), oneshows that the maps ∇ j induce a family of commuting ∗ -derivations on the linking algebra via∆ j (cid:18) a ξη ∗ b (cid:19) := (cid:18) ∂ Lj a ∇ j ( ξ ) −∇ j ( η ) ∗ ∂ Rj b (cid:19) , and the norms k e k n defined above are the restrictions of the norms obtained from the derivations∆ j . This proves that E n is an A n - B n bimodule. Since E n is a subspace of E , the propertiesof Definition 2.6 will follow once we show that the left and right inner products on A E B takevalues in A n and B n , respectively, when restricted to E n . This in turn follows since the innerproducts are realised as multiplications in the linking algebra. Lastly we have k e k k,ω ≤ k e k k forany state ω , so the image of E n is finite in each of the seminorms k e k k,ω with k ≤ n . (cid:3) Remark . The pre-Morita equivalence bimodule A ∞ E ∞B ∞ is the ‘smoothest’ bimodule overpre- C ∗ -algebras that we can consider from the derivations { ∂ Lj } dj =1 and { ∂ Rj } dj =1 . The lower-order pre-Morita equivalence bimodules A n E n B n for 1 ≤ n < ∞ will allow us to consider C ∗ -module and Hilbert space frames of differing regularity. This will be of use to us whenconsidering the localisation dichotomy of Wannier bases in Section 5.3. .4. Groupoid C ∗ -algebras and equivalence. Groupoids provide a useful generalisation ofgroups and group actions on spaces. A standard reference for groupoid C ∗ -algebras is [30]. Definition 2.15.
A groupoid is a set G with a subset G (2) ⊂ G × G , a multiplication map G (2) → G , ( γ, ξ ) γξ and an inverse G → G γ γ − such that(i) ( γ − ) − = γ for all γ ∈ G ,(ii) if ( γ, ξ ) , ( ξ, η ) ∈ G (2) , then ( γξ, η ) , ( γ, ξη ) ∈ G (2) ,(iii) ( γ, γ − ) ∈ G (2) for all γ ∈ G ,(iv) for all ( γ, ξ ) ∈ G (2) , ( γξ ) ξ − = γ and γ − ( γξ ) = ξ .Given a groupoid we denote by G (0) = { γγ − : γ ∈ G} the space of units and define the sourceand range maps r, s : G → G (0) by the equations r ( γ ) = γγ − , s ( γ ) = γ − γ for all γ ∈ G . The source and range maps allow us to characterise G (2) = (cid:8) ( γ, ξ ) ∈ G × G : s ( γ ) = r ( ξ ) (cid:9) . Throughout this work, we will assume that G is equipped with second countable, locally compactand Hausdorff topology such that the mulitplication, inversion, source and range maps are allcontinuous. A groupoid G is ´etale if the range map r : G → G (0) is a local homeomorphism.´Etale groupoids have the useful property that for all x ∈ G (0) , the fibres r − ( x ) and s − ( x ) arediscrete. Examples . (i) Let G be a group, then it is also a groupoid such that G (0) = { e } withmultiplication and inverse given by the group operation. If G is discrete, then it is ´etaleas a groupoid.(ii) Let X be a locally compact Hausdorff space, G a locally compact group and supposethere is a continuous left-action G → Homeo( X ) so that ( g, x ) g · x is jointly contin-uous. We can define the locally compact and Hausdorff transformation groupoid X ⋊ G given by pairs ( x, g ) ∈ X × G such that ( X ⋊ G ) (0) = X ,( x, g ) − = ( g − · x, g − ) , ( x, g )( g − · x, h ) = ( x, gh ) , s ( x, g ) = g − · x, r ( x, g ) = x. Definition 2.17.
Let G be a locally compact and Hausdorff groupoid. A continuous map σ : G (2) → T is a 2-cocycle if σ ( γ, ξ ) σ ( γξ, η ) = σ ( γ, ξη ) σ ( ξ, η )for any ( γ, ξ ) , ( ξ, η ) ∈ G (2) , and σ ( γ, s ( γ )) = 1 = σ ( r ( γ ) , γ )for all γ ∈ G . We will call a groupoid 2-cocycle normalised if σ ( γ, γ − ) = 1 for all γ ∈ G . Lemma 2.18.
If a groupoid -cocycle σ : G (2) → T is normalised, then σ ( γ, ξ ) = σ ( γξ, ξ − ) and σ ( γ, ξ ) = σ ( ξ − , γ − ) for all ( γ, ξ ) ∈ G (2) .Proof. Using the 2-cocyle identity, σ ( γ, ξ ) σ ( γξ, ξ − ) = σ ( γ, s ( ξ − )) σ ( ξ, ξ − ) = 1so σ ( γ, ξ ) = σ ( γξ, ξ − ). Next we compute that σ ( γξ, ξ − ) σ ( γξξ − , γ − ) = σ ( γξ, ξ − γ − ) σ ( ξ − , γ − ) = σ ( ξ − , γ − )and using the first identity σ ( γξ, ξ − ) σ ( γξξ − , γ − ) = σ ( γ, ξ ) σ ( γξξ − γ − , γ ) = σ ( γ, ξ ) σ ( r ( γξ ) , γ ) = σ ( γ, ξ )which gives that σ ( γ, ξ ) = σ ( ξ − , γ − ). (cid:3) We briefly review the construction of groupoid C ∗ -algebras. efinition 2.19. A Haar system on a locally compact Hausdorff groupoid G is a set of measures { ν x : x ∈ G (0) } on G such that supp( ν x ) = r − ( x ) and for all f ∈ C c ( G ), Z G f ( ξ ) d ν r ( η ) ( ξ ) = Z G f ( ηξ ) d ν s ( η ) ( ξ ) , g ( x ) := Z G f ( ξ ) d ν x ( ξ ) ∈ C ( G (0) ) . ´Etale groupoids always have a Haar system given by the counting measure on the (discrete)fibres r − ( x ). Given G with a 2-cocycle σ and Haar system { ν x } x ∈G (0) , we can define a ∗ -algebrastructure on C c ( G , σ ),( f ∗ f )( η ) = Z G f ( ξ ) g ( ξ − η ) σ ( ξ, ξ − η ) d ν r ( η ) ( ξ ) , f ∗ ( ξ ) = σ ( ξ, ξ − ) f ( ξ − ) . We will restrict ourselves to normalised cocycles, σ ( ξ, ξ − ) = 1 as it covers all examples ofinterest to us. See [30] for the general construction. The algebra C c ( G , σ ) has a right C ( G (0) )-module structure, where ( f · g )( ξ ) = f ( ξ ) g ( s ( ξ )) for f ∈ C c ( G , σ ) and g ∈ C ( G (0) ). We candefine a C ( G (0) )-valued inner-product( f | f ) G (0) ( x ) = Z G f ( ξ − ) f ( ξ − ) σ ( ξ, ξ − ) d ν x ( ξ ) = Z G f ( ξ − ) f ( ξ − ) d ν x ( ξ )as σ is normalised. Completing this space in the norm of C ( G (0) ) gives a right- C ( G (0) )-module,which we denote by L ( G , ν ) G (0) . There is a canonical left-action of C c ( G , σ ) on L ( G , ν ) G (0) bythe (twisted) convolution product. Definition 2.20 (cf. [19]) . The reduced groupoid C ∗ -algebra C ∗ r ( G , σ ) is the completion of C c ( G , σ ) in the norm inherited from the embedding C c ( G , σ ) ֒ → End ∗ C ( G (0) ) ( L ( G , ν )).If there is a topological space Z with (continuous) map ρ : Z → G (0) , we denote by G ⋉ ρ Z and Z ⋊ ρ G the pullback with respect to the source and range maps respectively. Definition 2.21 ( G -space) . A Hausdorff topological space Z is a left G -space if there is acontinuous map, called the anchor or moment map, ρ : Z → G (0) and a continuous map G ⋉ ρ Z → Z, ( γ, z ) γ · z ∈ Z such that for ( γ, z ) ∈ G ⋉ ρ Z and ( γ , γ ) ∈ G (2) , ρ ( γ · z ) = r ( γ ) , ρ ( z ) · z = z, ( γ γ ) · z = γ · ( γ · z )Unless otherwise stated, we will always assume that the moment map ρ : Z → G (0) is open andsurjective. One may also consider a right H -space from φ : Z → H (0) , where the definition isanalogous to the above but instead using a map Z ⋊ φ H → Z such that ρ ( z · η ) = s ( η ). Whenthe context is clear, we will write left/right-actions as γz or zη .We say that Z is a proper G -space (or that G acts properly) if the map G ⋉ ρ Z → Z × Z, ( γ, z ) ( γ · z, z )is proper. If the map ( γ, z ) ( γ · z, z ) is injective, then we say that Z is a free G -space or that G acts freely. Definition 2.22.
Let G and H be (locally compact, Hausdorff) groupoids and assume that Z carries both a left- G and right- H action via moment maps ρ : Z → G (0) and φ : Z → H (0) . Wesay that Z is a G – H -bibundle if the actions commute, i.e.,(1) for all ( γ, z ) ∈ G ⋉ ρ Z and ( z, η ) ∈ Z ⋊ φ H , ( γ · z ) · η = γ · ( z · η ),(2) for all ( z, η ) ∈ Z ⋊ φ H , ρ ( z · η ) = ρ ( z ),(3) for all ( γ, z ) ∈ G ⋉ ρ Z , φ ( γ · z ) = φ ( z ). G – H bibundle is a groupoid equivalence if the maps G ⋉ ρ Z → Z ∗ H (0) Z, ( γ, z ) ( γ · z, z ) ,Z ⋊ φ H → Z ∗ G (0) Z, ( z, η ) ( z, z · η )are homeomorphisms.At the level of operator algebras, groupoid equivalence implies Morita equivalence of the (fullor reduced) groupoid C ∗ -algebras [28, 32]. Let G ρ ←− Z φ −→ H be a groupoid equivalence suchthat G and H have Haar systems { ν x } x ∈G (0) and { λ y } y ∈H (0) respectively. Then there is a left(resp. right) action of C c ( G ) (resp. C c ( H )) on C c ( Z ) given by( f · ξ )( z ) = Z G f ( γ ) ξ ( γ − z ) d ν ρ ( z ) ( γ ) , ξ ∈ C c ( Z ) , f ∈ C c ( G ) , ( ξ · g )( z ) = Z H ξ ( zη ) g ( η − ) d λ φ ( z ) ( η ) , ξ ∈ C c ( Z ) , g ∈ C c ( H ) . (6)There is also the C c ( H )-valued inner product(7) ( ξ | ξ ) H ( η ) = Z G ξ ( γ − z ) ξ ( γ − zη ) d ν ρ ( z ) ( γ )where z ∈ Z is chosen such that φ ( z ) = r ( η ). Proposition 2.23 ([32], Theorem 4.1) . Let Z be a G – H groupoid equivalence. Then C c ( Z ) isa pre-Morita equivalence bimodule for C c ( G ) and C c ( H ) . Consequently, C ∗ r ( G ) and C ∗ r ( H ) areMorita equivalent. We denote by G L ( Z ) H the Morita equivalence bimodule that links C ∗ r ( G ) and C ∗ r ( H ). Becausewe work with reduced groupoid C ∗ -algebras, the Morita equivalence bimodule completion of C c ( Z ) is constructed from the linking groupoid L = G ⊔ Z ⊔ Z op ⊔ H [32]. In Section 4.1 wediscuss instances of twisted groupoid Morita equivalence.3. Gabor frames and Wannier bases from groupoid equivalences
We let G ρ ←− Z φ −→ H be a G – H equivalence of locally compact, second countable and Hausdorffgroupoids such that H (0) is compact and H is ´etale. In particular, this implies that C ∗ r ( H ) isunital. We assume that G has a Haar system { ν x } x ∈G (0) and so C c ( G ) C c ( Z ) C c ( H ) is a pre-Moritaequivalence bimodule described by Equations (6) and (7). Because H is ´etale, the right-actionin Equation (6) reduces to a sum over the discrete set r − ( φ ( z )).3.1. C ∗ -module frame. Because all C ∗ -algebras are separable, there exists a countable rightmodule frame for G L ( Z ) H . Furthermore, since C ∗ r ( H ) is unital, G L ( Z ) H is finitely generatedand projective as a left C ∗ r ( G )-module and so has a finite left C ∗ -module frame.A right module frame for the submodule C c ( Z ) is constructed in [28, Proposition 2.10]. Webriefly review this construction. We say that a subset L ⊂ G is r -relatively compact if L ∩ r − ( K )is relatively compact for every compact K ⊂ G . We consider a triple ( K, U, ε ) with K ⊂ G (0) compact with U ⊂ G an r -relatively compact neighbourhood of G (0) and ε >
0. Because G (0) is paracompact and G acts properly on Z , there are open, relatively compact sets { V j } nj =1 ⊂ Z such that { ρ ( V j ) } nj =1 cover K and are such that ( γz, z ) ∈ V j × V j implies that γ ∈ U . We takea partition of unity { ψ j } nj =1 subordinate to the cover { ρ ( V j ) } Nj =1 . We can then find functions { ˜ ψ j } nj =1 such that supp( ˜ ψ j ) ⊂ V j and X η ∈ r − ( φ ( z )) ˜ ψ j ( zη ) = ψ j ( ρ ( z )) . inally, we can approximate ˜ ψ j with { ϕ j } nj =1 such that supp( ϕ j ) ⊂ V j and (cid:12)(cid:12)(cid:12) ˜ ψ j ( z ) − ϕ j ( z ) Z G ϕ j ( γ − z ) d ν ρ ( z ) ( γ ) (cid:12)(cid:12)(cid:12) ≤ εM M = sup z n X j =1 X η ∈ r − ( φ ( z )) χ V j ( zη ) . The functions e n = P nj =1 G ( ϕ j | ϕ j ) are an approximate identity for the left action of C c ( G ) on C c ( Z ). The functions { ϕ j } ⊂ C c ( Z ) can then be used to construct a right C ∗ -module frame inthe Morita equivalence bimodule G L ( Z ) H [32].3.2. Localisation and Hilbert space frame.
For any x ∈ H (0) we have a state ω x given bythe restriction of f ∈ C c ( H ) to H (0) and then evaluation at x ∈ H (0) . We see that ω x ( g ∗ g ) = X η ∈ r − ( x ) | g ( η ) | , which shows that ω x is positive. There are other possible states on C ∗ r ( H ) that one may alsoconsider such as integration with respect to a quasi-invariant measure [30]. The reason we choosethe evaluation state is because we would like to construct Gabor frames that have a discretelabeling. By choosing x ∈ H (0) , the discrete set r − ( x ) provides us with such a labeling. Lemma 3.1.
Fix an element x ∈ H (0) . Then there are real-valued functions { δ α } α ∈ r − ( x ) ⊂ C c ( H ) such that ω x ( δ α ∗ δ ∗ α ) = δ α ,α .Proof. Because H (0) is compact and r − ( x ) is discrete, for each α ∈ r − ( x ) we take δ α a bumpfunction supported on U α such that U α ∩ r − ( x ) = { α } and δ α ( α ) = 1. Then we compute that ω x (cid:0) δ α ∗ δ ∗ α (cid:1) = X β ∈ r − ( x ) δ α ( β ) δ α ( β ) = δ α ( α ) = δ α ,α as the sum vanishes everywhere except for at most one term. (cid:3) Lemma 3.2.
Let Z x = φ − ( x ) and for z ∈ Z x define the measure ν ρ ( z ) on Z x by Z Z x f ( w )d ν ρ ( z ) ( w ) := Z G f ( γ − · z )d ν ρ ( z ) ( γ ) . The Hilbert space localisation h x of G L ( Z ) H in ω x is L ( Z x , d ν ρ ( z ) ) with z ∈ Z x chosen arbi-trarily.Proof. We consider the inner-product, where h e , e i x = ω x (cid:0) ( e | e ) H (cid:1) = ( e | e ) H ( x ) = Z G e ( γ − z ) e ( γ − zx ) d ν ρ ( z ) ( γ )= Z G e ( γ − z ) e ( γ − zφ ( z )) d ν ρ ( z ) ( γ ) = Z G e ( γ − z ) e ( γ − z ) d ν ρ ( z ) ( γ )Hence the Hilbert space is the L -completion of C c ( Z x ) with respect to the measure ν ρ ( z ) . Since Z is an equivalence, for every z, w ∈ Z x there exists γ ∈ r − ( ρ ( z )) ⊂ G such that w = γ − z .The measure ν ρ ( z ) is thus independent of the choice of z ∈ Z x . (cid:3) Given x ∈ H (0) and e ∈ G L ( Z ) H , we let e x be the corresponding element in the localisation L ( Z x ). Given any α ∈ r − ( x ), define a function e αx ∈ L ( Z x ) by ( e αx )( y ) = e ( yα ), y ∈ Z x . Lemma 3.3.
Let e ∈ G L ( Z ) H , a ∈ H (0) and α ∈ r − ( x ) . (i) There is an equality e αx = ( e · δ ∗ α ) x with δ α be the bump functions from Lemma 3.1. (ii) If e ∈ G L ( Z ) H is such that ( e | e ) H = 1 H , then { e αx } α ∈ r − ( x ) is an orthonormal systemin L ( Z x ) ; iii) Given e , e , ξ ∈ G L ( Z ) H , (cid:0) Θ Re ,e ( ξ ) (cid:1) x = X α ∈ r − ( x ) ( e ) αx h ( e ) αx , ξ x i x . Proof.
We first note that e ( yα ) is well-defined as ρ ( y ) = x = r ( α ). For part (i) we compute for y ∈ Z x , ( e · δ ∗ α )( y ) = X β ∈ r − ( x ) e ( yβ ) δ ∗ α ( β − ) = X β ∈ r − ( x ) e ( yβ ) δ α ( β ) = e ( yα ) . Using part (i) and Lemma 3.1, we see that for e such that ( e | e ) H = 1, h e α x , e α x i x = ω x (cid:0) ( e · δ ∗ α | e · δ ∗ α ) H (cid:1) = ω x (cid:0) δ α ( e | e ) H δ ∗ α (cid:1) = δ α ,α and so { e αx } α ∈ r − ( x ) is an orthonormal system, which proves part (ii).For part (iii), we again compute for y ∈ φ − ( x ) (cid:0) Θ Re ,e ( ξ ) (cid:1) ( y ) = X α ∈ r − ( x ) e ( yα )( e | ξ ) H ( α − )= X α ∈ r − ( x ) e ( yα ) Z G e ( γ − z ) ξ ( γ − zα − ) d ν ρ ( z ) ( γ ) . We now let u = zα − , where uα = zs ( α ) = zφ ( z ) = z and ρ ( u ) = ρ ( zα − ) = ρ ( z ) as Z is agroupoid equivalence. Hence (cid:0) Θ Re ,e ( ξ ) (cid:1) ( y ) = X α ∈ r − ( x ) e ( yα ) Z G e ( γ − uα ) ξ ( γ − u ) d ν ρ ( u ) ( γ )= X α ∈ r − ( x ) ( e ) αx ( y ) h ( e ) αx , ξ x i x . (cid:3) For a countable set J and a C ∗ -algebra B we denote by ℓ ( J, B ) the standard Hilbert C ∗ -moduleof sequences indexed by J . That is ℓ ( J, B ) := n f : J → B : X j ∈ J f ( j ) ∗ f ( j ) < ∞ o , where the series converges in B . Theorem 3.4.
Let { e j } j ∈ J ⊂ G L ( Z ) H be a countable subset and E := span H { e j : j ∈ J } theclosed C ∗ r ( H ) submodule generated by { e j } j ∈ J . The following are equivalent: (1) The sequence { e j } j ∈ J is a right C ∗ -module frame of E ⊂ G L ( Z ) H ; (2) For ξ ∈ E the map j ( e j | ξ ) H takes values in ℓ ( J, C ∗ r ( H )) and for all x ∈ H (0) theset { ( e j ) αx } j ∈ J,α ∈ r − ( x ) is a normalised tight frame for E x ⊂ L ( Z x ) ;Proof. (1) ⇒ (2): Using part (iii) of Lemma 3.3, we see that X j ∈ J X α ∈ r − ( x ) ( e j ) αx h ( e j ) αx , ξ x i x = X j ∈ J (cid:0) Θ Re j ,e j ( ξ ) (cid:1) x = ξ x as ( e j ) j ∈ J is a right C ∗ -module frame.(2) ⇒ (1): In order to prove that { e j } j ∈ J is a Hilbert C ∗ -module frame for E , we need to showthat the map v : E → ℓ ( J, C ∗ r ( H )) , ξ ( e j | ξ ) H , satisfies v ∗ v = 1. Note that v is well-defined since we assume that j ( e j | ξ ) H is an elementof ℓ ( J, C ∗ r ( H )), and v is automatically adjointable with v ∗ ( b j ) = P j e j · b j . ince { ( e j ) αx } j ∈ J,α ∈ r − ( x ) is a normalised tight frame for E x ⊂ L ( Z x ), the map v x : L ( Z x ) → ℓ ( J × r − ( x )) , ψ
7→ h ( e j ) αx , ψ i x , is an isometry, for( ψ | ψ ) x = k ψ k x = X ( j,α ) ∈ J × r − ( x ) (cid:12)(cid:12) h ( e j ) αx , ψ i x (cid:12)(cid:12) = k v x ( ψ ) k = h v x ψ, v x ψ i ℓ ( J × r − ( x )) , and it follows that v ∗ x v x = 1 . The map v ∗ x is given by v ∗ x : ( λ αj ) X j, α ( e j ) αx λ αj , and we find from Lemma 3.3 ξ x = v ∗ x v x ( ψ ) = X α,j ( e j ) αx h ( e j ) αx , ξ x i x = (cid:16) X j Θ e j ,e j ( ξ ) (cid:17) x = ( v ∗ v ( ξ )) x . Therefore we can conclude that k ξ − v ∗ v ( ξ ) k = sup x k ξ − v ∗ v ( ξ ) k x = sup x k ξ x − ( v ∗ v ( ξ )) x k x = 0 , and hence v ∗ v = 1. (cid:3) Remark . Well-definedness of the map v : E → ℓ ( J, C ∗ r ( H )) entails that for all e ∈ E theseries X j ( e | e j ) H ( e j | e ) H , is norm convergent in C ∗ r ( H ). This is automatic when the index set J is finite but poses anon-trivial restriction for infinite J .3.3. Finitely generated and projective modules.
Using the left-action of C ∗ r ( G ) on L ( Z ),we can define a representation of π x : C ∗ r ( G ) → B ( h x ), where π x ( f ) ξ x = ( f · ξ ) x , f ∈ C ∗ r ( G ) , ξ ∈ L ( Z ) . Concretely,( π x ( f ) ξ x )( y ) = Z G f ( γ ) ξ ( γ − y ) d ν ρ ◦ φ − ( x ) ( γ ) , f ∈ C c ( G ) , ξ ∈ C c ( Z ) . We consider projections, p = p ∗ = p ∈ M n ( C ∗ r ( G )), which act compactly on L ( Z ) ⊕ n . Proposition 3.6.
Let p = p ∗ = p ∈ M n ( C ∗ r ( G )) . There is a finite set { v j } nj =1 ⊂ pL ( Z ) ⊕ n such that for any x ∈ H (0) , { v α , . . . , v αn } α ∈ r − ( x ) is a normalised tight frame of π x ( p ) h ⊕ nx .Proof. By Lemma 2.9 (2), there is a finite frame { v j } nj =1 of pL ( Z ) ⊕ n . It is immediate thatthe localisation of pL ( Z ) ⊕ n in ω x is π x ( p ) h ⊕ nx . The result then follows by the same proof asTheorem 3.4. (cid:3) Let us now consider the converse, i.e. given the Hilbert space frames { w α , . . . , w αm } α ∈ r − ( x ) for x ∈ H (0) , we construct a finitely generated and projective module. Proposition 3.7.
Let W ⊂ G L ( Z ) H be a closed submodule and { w , · · · , w n } a finite sub-set of G L ( Z ) H such that W := span C ∗ r ( H ) { w , · · · , w n } . Suppose that for all x ∈ H (0) , { w α , . . . , w αm } α ∈ r − ( x ) is a normalised tight frame of W x ⊂ L ( Z x , d ν ρ ( x ) ) . Then W is a finitelygenerated and projective module over C ∗ r ( H ) . If, for each x ∈ H (0) , { w α , . . . , w αm } α ∈ r − ( x ) is anorthonormal basis, then W ∼ = C ∗ r ( H ) ⊕ m . roof. The first part of the Proposition follows immediately from Theorem 3.4 and the factthat any C ∗ r ( H )-module with a finite C ∗ -module frame is finitely generated and projective. Nowsuppose that { w α , . . . , w αm } α ∈ r − ( x ) is an orthonormal basis and let p jk = ( w j | w k ) H ∈ C ∗ r ( H ).For any x ∈ H (0) , we have that δ j,k δ α ,α = h w α j , w α k i x = ω x (cid:0) δ α ( w j | w k ) H δ ∗ α (cid:1) = X β ∈ r − ( x ) ( δ α ∗ p jk )( β ) δ α ( β )= ( δ α ∗ p jk )( α ) = X η ∈ r − ( x ) δ α ( η ) p jk ( η − α ) = p jk ( α − α )for all α , α ∈ r − ( x ) and all j, k ∈ { , . . . , m } . Now, for any η ∈ H , we can find some x ∈ H (0) and α, β ∈ r − ( x ) such that η = α − β . Hence, p jk ( η ) = p jk ( α − β ) = δ j,k δ α,β . This impliesthat the matrix p ∈ M m ( C ∗ r ( H )) is the identity matrix. Hence W ∼ −→ C ∗ r ( H ) ⊕ m . (cid:3) Theorem 3.8.
Let p = p ∗ = p ∈ M n ( C ∗ r ( G )) . The finitely generated and projective module pL ( Z ) ⊕ n with frame { v j } mj =1 is isomorphic to the free module C ∗ r ( H ) ⊕ m if and only if for all x ∈ H (0) , { v α , . . . , v αm } α ∈ r − ( x ) is an orthonormal basis of π x ( p ) h ⊕ nx .Proof. Suppose there is a unitary isomorphism of C ∗ -modules ϕ : pL ( Z ) ⊕ n ∼ −→ C ∗ r ( H ) ⊕ m . It isclear that { j } mj =1 is right-frame of C ∗ r ( H ) ⊕ m and because ϕ respects the inner-product structure { v j } mj =1 is a frame of pL ( Z ) ⊕ n with v j = ϕ − (1 j ). We know that { v αj } is a normalised tightframe by Proposition 3.6 and we see that h ( v j ) α x , ( v k ) α x i x = ω x (cid:0) ( v j · δ ∗ α | v k · δ ∗ α ) H (cid:1) = ω x (cid:0) δ α ( ϕ − (1 k ) | ϕ − (1 j )) H δ ∗ α (cid:1) = δ j,k ω x ( δ α ∗ δ ∗ α ) = δ j,k δ α ,α . Hence the tight frame is orthonormal and so is an orthonormal basis. The converse statementfollows from Proposition 3.7. (cid:3)
Remark K -theoretic interpretation) . Theorem 3.8 has an interpretation via the K -theory ofthe groupoid C ∗ -algebras. Using the ∗ -isomorphism C ∗ r ( G ) ∼ = K C ∗ r ( H ) ( L ( Z )), we can naturallyconsider any projection p ∈ M n ( C ∗ r ( G )) as a finite-rank operator on L ( Z ) ⊕ n H . Therefore we canalso consider p as a projection in M m ( C ∗ r ( H )) for some m , with corresponding K -theory class[ p ] ∈ K ( C ∗ r ( H )). If pL ( Z ) H ∼ = C ∗ r ( H ) ⊕ m , then [ p ] = m [1] ∈ K ( C ∗ r ( H )) and the projection p is trivial in reduced K -theory.Let us now consider the case where there are dense pre- C ∗ -algebras A ⊂ C ∗ r ( G ) and B ⊂ C ∗ r ( H ),which will allow us to consider C ∗ -modules and Hilbert space frames with additional regularityas in Proposition 2.13. Thus suppose there are families of commuting ∗ -derivations { ∂ H j } dj =1 : C c ( H ) → C c ( H ) , { ∂ G j } dj =1 : C c ( G ) → C c ( G ) , as well as a family of maps ∇ j : C c ( Z ) → C c ( Z ) such that for each j and all a ∈ C c ( G ) , ξ, η ∈ C c ( Z ), Equations (3) and (4) hold. Denote by S k ( G ) and S k ( H ) the degree k Fr´echet completionsof C c ( G ) and C c ( H ) in these seminorms (Proposition 2.10), and by S k ( Z ) the degree k Fr´echetcompletion of C c ( Z ). We often write S for S ∞ in each of these cases. By Proposition 2.13, S k ( G ) S k ( Z ) S k ( H ) is a pre-Morita equivalence bimodule for all k = 0 , · · · , ∞ . Theorem 3.10.
Let k = 0 , · · · , ∞ and p = p ∗ = p ∈ M n ( S k ( G )) . Then there is a finite frame { v j } mj =1 of p S k ( Z ) ⊕ n S k ( H ) such that for all x ∈ H (0) the normalised tight frame { v β , . . . , v βm } β ∈ r − ( x ) of π x ( p ) h ⊕ nx is finite with respect to the seminorms k · k l,x on h x for ≤ l ≤ k . There is anisomorphism p S k ( Z ) ⊕ n S k ( H ) ∼ = S k ( H ) ⊕ m if and only if for all x ∈ H (0) , { v β , . . . , v βm } β ∈ r − ( x ) is anorthonormal basis. roof. Because v j · δ ∗ β ⊂ S k ( Z ) for any x ∈ H (0) and β ∈ r − ( x ), the first statement thenfollows from Lemma 2.9 and Proposition 2.13. Lemma 2.9 and the fact that S k ( H ) is a pre- C ∗ -algebra give that p S k ( H ) ⊕ n is a free S k ( H )-module if and only if its C ∗ -completion is a free C ∗ r ( H )-module.If p S k ( Z ) ⊕ n S k ( H ) ∼ = S k ( H ) ⊕ m , then there is a finite frame { v , . . . , v m } ⊂ p S k ( Z ) ⊕ n such that( v i | v j ) B = δ i,j S k ( H ) . By part (ii) of Lemma 3.3 we therefore see that { v , . . . , v αm } α ∈ r − ( x ) is an orthonormal system and hence must be an orthonormal basis. For the converse, we canuse Proposition 3.7 and the fact that the finite C ∗ r ( H )-module frame can be approximatedarbitrarily well by a finite S k ( H )-module frame of the same size. (cid:3) Gabor frames and Wannier bases from twisted transversals
Here we consider our framework in the case of groupoid equivalences that come from abstracttransversals with an additional twist by a normalised groupoid 2-cocycle. In Section 5 we applythese results to the Delone transversal groupoid with twist coming from a magnetic field.4.1.
Twisted Morita equivalence.Definition 4.1.
A topological groupoid G admits an abstract transversal if there is a closedsubset X ⊂ G (0) such that(i) X meets every orbit of the G -action on G (0) ;(ii) for the relative topologies on X and G X := { γ ∈ G : s ( γ ) ∈ X } ⊂ G , the restrictions r : G X → G (0) and s : G X → X are open maps.Given an abstract transversal X ⊂ G (0) , G r ←− G X s −→ H is a G – H groupoid equivalence for H = { γ ∈ G X : r ( γ ) ∈ X } , see [28, Example 2.7]. Examples of abstract transversals includetransitive groupoids and groupoids from foliations.We now fix a locally compact, second countable and Hausdorff groupoid G such that X ⊂ G (0) iscompact and admits an abstract transversal G X with H ´etale. We also fix a normalised groupoid2-cocycle σ G on G , i.e. σ G ( γ, γ − ) = 1 for all γ ∈ G . The restriction of σ G then gives a groupoid2-cocycle σ H for H . The 2-cocycle twists the module structure( f · e )( z ) = Z G f ( γ ) e ( γ − z ) σ G ( γ, γ − z ) d ν r ( z ) ( γ ) , e ∈ C c ( G X ) , f ∈ C c ( G , σ G ) , ( e · g )( z ) = X η ∈ r − ( s ( z )) e ( zη ) g ( η − ) σ G ( zη, η − ) , e ∈ C c ( G X ) , g ∈ C c ( H , σ H ) , ( e | e ) H ( η ) = Z G e ( γ − z ) e ( γ − zη ) σ G ( z − γ, γ − zη ) d ν r ( z ) ( γ ) , r ( η ) = s ( z ) . Proposition 2.23 can be extended to the case of such simple twists. The case of general groupoidtwists arising from S -extensions is handled via equivalence of Fell bundles, see [27, 33]. Proposition 4.2.
The module C c ( G X ) is a pre-Morita equivalence bimodule for C c ( G , σ G ) and C c ( H , σ H ) . Consequently, C ∗ r ( G , σ G ) and C ∗ r ( H , σ H ) are Morita equivalent. Twisted Gabor frames and Wannier bases.
We now outline the minor changes re-quired to recover the results of Section 3 to the case of twisted algebras. One advantage ofrestricting to normalised cocycles is that in the Hilbert space localisation, the inner-product implifies. Namely, for x ∈ X = H (0) , h ( e ) x , ( e ) x i x = ( e | e ) H ( x ) = Z G e ( γ − z ) e ( γ − z ) σ G ( z − γ, γ − z ) d ν r ◦ s − ( x ) ( γ )= Z G e ( γ − z ) e ( γ − z ) d ν r ◦ s − ( x ) ( γ ) . Lemma 4.3.
Let x ∈ X and α ∈ r − ( x ) . For e x ∈ h x , define e αx ( y ) = e ( yα ) σ G ( yα, α − ) . (i) If e ∈ L ( G X , σ ) is such that ( e | e ) H = 1 H , then { e αx } α ∈ r − ( x ) is an orthonormal systemin h x . (ii) Given e , e , ξ ∈ G L ( G X , σ ) H , (cid:0) Θ Re ,e ( ξ ) (cid:1) x = X α ∈ r − ( x ) ( e ) αx h ξ x , ( e ) αx i x Proof.
Like in the untwisted case, we see that for y ∈ s − ( x ),( e · δ ∗ α )( y ) = X β ∈ r − ( x ) e ( yβ ) δ α ( β ) σ G ( yβ, β − ) = e ( yα ) σ G ( yα, α − ) . Hence we can compute h e α x , e α x i x = ω x ( δ α ∗ δ ∗ α ) = X β ∈ r − ( x ) δ α ( β ) δ α ( β ) σ H ( β, β − ) = δ α ,α . For part (ii), can follow the same argument as Lemma 3.3. For y ∈ s − ( x ) and u = zα − with uα = zs ( α ) = zs ( z ) = z and r ( u ) = r ( zα − ) = r ( z ), (cid:0) Θ Re ,e ( ξ ) (cid:1) ( y ) = X α ∈ r − ( x ) e ( yα ) σ G ( yα, α − ) Z G e ( γ − uα ) ξ ( γ − u ) σ G ( α − u − γ, γ − u ) d ν r ( u ) ( γ )= X α ∈ r − ( x ) e ( yα ) σ G ( yα, α − ) Z G e ( γ − uα ) σ G ( γ − uα, α − ) ξ ( γ − u ) d ν r ( u ) ( γ )= X α ∈ r − ( x ) ( e ) αx ( y ) h ( e ) αx , ξ x i x , where we used the 2-cocycle identity σ G ( α − , u − γ ) σ G ( α − u − γ, γ − u ) = σ G ( α − , s ( γ − u )) σ G ( u − γ, γ − u ) = 1which implies that σ G ( α − u − γ, γ − u ) = σ G ( α − , u − γ ). Then using Lemma 2.18 σ G ( α − , u − γ ) = σ G ( γ − u, α ) = σ G ( γ − uα, α − ) . (cid:3) Given elements w α j , w α k ∈ h x = L ( G X , d ν r ◦ s − ( x ) ), we can compute that h w α j , w α k i x = ω x (cid:0) δ α ( w j | w k ) H δ ∗ α (cid:1) = (cid:0) δ α ∗ ( w j | w k ) H (cid:1) ( α )= ( w j | w k ) H ( α − α ) σ H ( α , α − α ) . Hence, if h w α j , w α k i x = δ j,k δ α ,α for some α , α ∈ r − ( x ), the 2-cocycle term will be 1.At this point we can follow the same arguments as those in Theorem 3.4 and Section 3.3, so wesummarise our results. Proposition 4.4. If ( e j ) j ∈ J is a right C ∗ -module frame of G L ( G X , σ ) H , then for all x ∈ X the set { ( e j ) αx } for j ∈ J and α ∈ r − ( x ) is a normalised tight frame of h x . We can again define a representation of π x : C ∗ r ( G , σ G ) → B ( h x ), where( π x ( f ) ξ x )( y ) = Z G f ( γ ) ξ ( γ − y ) σ G ( γ, γ − y ) d ν r ◦ s − ( x ) ( γ ) , f ∈ C c ( G ) , ξ ∈ C c ( G X ) . e consider the case of pre- C ∗ -algebras S k ( G , σ ) ⊂ C ∗ r ( G , σ G ) and S k ( H , σ ) ⊂ C ∗ r ( H , σ H ) definedfrom families of derivations with a pre-Morita equivalence bimodule S k ( G ,σ ) S k ( G X , σ ) S k ( H ,σ ) defined from a family of maps ∇ j : C c ( Z ) → C c ( Z ) using the construction in Proposition 2.13. Proposition 4.5.
Let k = 0 , · · · , ∞ and p = p ∗ = p ∈ M n ( S k ( G , σ )) . There are elements { v j } mj =1 ⊂ p S k ( G X , σ ) ⊕ n S k ( H ,σ ) such that for all x ∈ X , { v β , . . . , v βm } β ∈ r − ( x ) is a normalised tightframe of π x ( p ) h ⊕ nx that is finite under the seminorms k · k l,x on h x for ≤ l ≤ k . This tightframe is an orthonormal basis for all x ∈ H (0) if and only if p S k ( G X , σ ) ⊕ n S k ( H ,σ ) ∼ = S k ( H , σ ) ⊕ m . Gabor frames and Wannier bases for the Delone groupoid
Delone sets and the transversal groupoid.
We review some of the material from [5]as outlined in [9]. We denote by B ( x ; K ) ⊂ R d the open ball centered at x with radius K > Definition 5.1.
Let
L ⊂ R d be discrete and infinite and fix 0 < r < R .(1) L is r -uniformly discrete if | B ( x ; r ) ∩ L| ≤ x ∈ R d .(2) L is R -relatively dense if | B ( x ; R ) ∩ L| ≥ x ∈ R d .An r -uniformly discrete and R -relatively dense set L is called an ( r, R )-Delone set. Proposition 5.2 ([13], Chapter 1) . The set of ( r, R ) -Delone sets is a compact and metrizablespace. Let d H denote the Hausdorff distance between sets. A neighbourhood base at L ∈
Del ( r,R ) is given by the sets U ε,M ( L ) = (cid:8) L ′ ∈ Del ( r,R ) : d H (cid:0) L ∩ B (0; M ) , L ′ ∩ B (0; M ) (cid:1) < ε (cid:9) , M, ε > . The set of Delone sets Del ( r,R ) is clearly invariant under translations and rotations. Definition 5.3.
Let Λ be a an ( r, R )-Delone subset of R d . The continuous hull of Λ is thedynamical system (Ω Λ , R d , T ), where Ω Λ ⊂ Del ( r,R ) is the closure of the orbit of Λ under thetranslation action.The continuous hull of Λ therefore gives a locally compact Hausdorff groupoid Ω Λ ⋊ R d . Thisgroupoid admits a transversal in the sense of Definition 4.1. Definition 5.4.
The transversal of Λ is given by the setΩ = {L ∈ Ω Λ : 0 ∈ L} , We see that Ω is a closed subset of Ω Λ and so is compact. Proposition 5.5 ([18], Lemma 2) . Given a Delone set Λ with transversal Ω , define the set G Del := (cid:8) ( L , x ) ∈ Ω × R d : x ∈ L (cid:9) , with maps ( L , x ) − = ( L − x, − x ) , ( L , x ) · ( L − x, y ) = ( L , x + y ) , s ( L , x ) = L − x, r ( L , x ) = L and unit space G (0) = Ω . Then G Del is a Hausdorff ´etale groupoid in the relative topologyinherited from Ω × R d .Notation. Following the previous proposition, we will let G Del denote the ´etale groupoid from aDelone set. We let F = Ω Λ ⋊ R d be the crossed product groupoid. Proposition 5.6 ([9], Proposition 2.16) . Let
L ⊂ R d be an ( r, R ) -Delone set with transversal Ω and associated groupoid G Del . For U ⊂ Ω an open set, the sets V ( U,y,ε ) := ( U × B ( y ; ε )) ∩ G Del = { ( L , x ) ∈ Ω × R d : L ∈
U, x ∈ L ∩ B ( y ; ε ) } , form a base for the topology on G Del . For < ε < r/ , the restriction s : V ( U,y,ε ) → Ω is ahomeomorphism onto its image. Moreover, the set Ω ⊂ Ω Λ is an abstract transversal and thegroupoid G Del ⊂ Ω Λ ⋊ R d , with the subspace topology, is equivalent to Ω Λ ⋊ R d . et us now fix a normalised 2-cocycle σ : (Ω Λ ⋊ R d ) (2) → T , σ ( γ, γ − ) = 1 for all γ ∈ Ω Λ ⋊ R d ,which also restricts to a 2-cocycle on G Del . Our main motivation to consider such twists comesfrom the following example.
Example . Working with the continuous hull Ω Λ ⋊ R d ,we can define a 2-cocycle, σ : F (2) → T as follows. We first define a parametrised magnetic fieldas a continuous map B : Ω Λ → V R d . Then we define σ (( L , x ) , ( L − x, y )) = exp (cid:0) − i Γ L h , x, x + y i (cid:1) , Γ L h x, y, z i = Z h x,y,z i B ( L )and h x, y, z i ⊂ R d is the triangle with corners x, y, z ∈ R d . Hence Γ L h , x, x + y i measures themagnetic flux through the triangle defined by the points 0 , x, x + y ∈ R d . It is shown in [6]that σ is a well-defined 2-cocycle. We remark that σ will always be trivial for d = 1 and isnormalised because σ (( L , x ) , ( L − x, − x )) = exp (cid:0) − i Γ h , x, i (cid:1) = 1 . If the magnetic field is constant over Ω Λ , then our general flux equation can be described usinga real-valued and skew-symmetric matrix B with σ (( L , x ) , ( L − x, y )) = exp (cid:0) − i h x, B ( x + y ) i (cid:1) = exp (cid:0) − i h x, By i (cid:1) . The 2-cocycle σ on Ω Λ ⋊ R d also restricts to a normalised 2-cocycle on G Del , where we notethat if (( L , x ) , ( L − x, y )) ∈ G (2)Del , the points 0 , x, x + y ∈ L and so Γ L h , x, x + y i gives a fluxthrough the triangle with points in L . Remark . Given a groupoid 2-cocycle σ : (Ω Λ ⋊ R d ) (2) → T , the twisted groupoid C ∗ -algebra C ∗ r (Ω Λ ⋊ R d , σ ) is canonically isomorphic to the twisted crossed product C (Ω Λ ) ⋊ σ ′ R d , where σ ′ : R d × R d → U ( C (Ω Λ )) , σ ′ ( x, y ) = σ (( L , x ) , ( L − x, y )) . Regularity, smoothness and decay properties of functions on Ω Λ ⋊ R d and G Del are encoded viathe groupoid 1-cocycles c k : Ω Λ ⋊ R d → R , c ( L , x ) = x k , k ∈ { , . . . , d } . and their restrictions to G Del . It is shown in [9, Proposition 2.17] that the groupoid cycles are exact , in that c − k (0) has a Haar system and c k is a quotient map onto its image.Given the cocycles c j : Ω Λ ⋊ R d → R d , we obtain families of commuting derivations { ∂ j } dj =1 onboth C c (Ω Λ ⋊ R d , σ ) and C c ( G Del , σ ) given by ( ∂ j f )( L , x ) = x j f ( L , x ). For k = 0 , · · · , ∞ , weobtain pre- C ∗ -algebra completions A k of C c (Ω Λ ⋊ R d , σ ) and B k of C c ( G Del , σ ) using Proposition2.10.5.2.
The transversal G Del -space and its localisation.
Following Section 4, we consider thespace F Ω := (cid:8) ( L , x ) ∈ Ω Λ ⋊ R d : x ∈ L (cid:9) , s : F Ω → Ω , s ( L , x ) = L − x, which implements a groupoid equivalence between F = Ω Λ ⋊ R d and G Del . Thus C c ( F Ω ) is apre-Morita equivalence bimodule for C c (Ω Λ ⋊ R d , σ ) and C c ( G Del , σ ) and can be completed intothe Morita equivalence bimodule F L ( F Ω , σ ) G Del between C ∗ r (Ω Λ ⋊ R d , σ ) and C ∗ r ( G Del , σ ). Lemma 5.9.
The restrictions of the cocycles c j : Ω Λ ⋊ R d → R to F Ω define maps ∇ j : C c ( F Ω ) → C c ( F Ω ) , ∇ j ( f )( L , x ) := x j f ( L , x ) . For all a ∈ C c ( F , σ ) , b ∈ C c ( G Del , σ ) and ξ, η ∈ C c ( F Ω ) we have ∇ j ( a · ξ · b ) = ∂ j ( a ) · ξ · b + a · ∇ j ( ξ ) · b + a · ξ · ∂ j ( b ) ,∂ j ( ξ | η ) G Del = ( ξ | ∇ j ( η )) G Del − ( ∇ j ( ξ ) | η ) G Del . onsequently, for all k = 0 , · · · , ∞ , the space C c ( F Ω ) can be completed into a pre-Moritaequivalence bimodule A k S k ( F Ω , σ ) B k for the pre- C ∗ -algebras A k ⊂ C ∗ r (Ω Λ ⋊ R d , σ ) and B k ⊂ C ∗ r ( G Del , σ ) .Proof. As F Ω and G Del are subspaces of Ω Λ ⋊ R d and the bimodule structure and inner productare induced by the convolution product in C c (Ω Λ ⋊ R d , σ ), the required identities follow fromthe fact that multiplication by x j is a derivation of C c (Ω Λ ⋊ R d , σ ). (cid:3) For every
L ∈ Ω , there is a state ω L on C ∗ r ( G Del , σ ) such that ω L ( f ) = f ( L ,
0) for f ∈ C c ( G Del , σ ). Note that ω L ( f ∗ f ) = X y ∈L | f ( L − y, − y ) | , ω L (1 C ∗ r ( G Del ) ) = 1and so ω L is faithful. From this point, all results from Section 4 apply to the Delone groupoidsetting. Though for concreteness, we highlight some key aspects of this example. Lemma 5.10.
For every
L ∈ Ω , the localised Hilbert space h L ∼ = L ( R d ) .Proof. We define a map β L → L ( R d ) that agrees with the localised Hilbert space inner product.Namely, we consider β L : L ( F Ω ) → L ( R d ), given by [ β L ( ξ )]( x ) = ξ ( L − x, − x ) for almost all x . To see why this is true, we first note that for any L ∈ Ω , s − ( L ) = { ( L − x, − x ) } x ∈ R d andthe measure on s − ( L ) is just the Lebesgue measure on R d . We also see that h ξ , ξ i L = ( ξ | ξ ) C ∗ r ( G Del ) ( L ,
0) = Z R d ξ ( L − y, − y ) ξ ( L − y, − y ) d y. Hence we see there is a canonical identification of h L with β L [ L ( F Ω )] ∼ = L ( R d ). (cid:3) For any ξ ∈ F L ( F Ω ) G Del , let ξ L ∈ h L be its localisation. Because ∇ α ξ ( L , x ) = x α ξ ( L , x ) for α ∈ N d and ξ ∈ C c ( F Ω ), Proposition 2.13 gives the following. Lemma 5.11.
For any k = 0 , · · · , ∞ and L ∈ Ω , every element in dense subspace β L [ S k ( F Ω )] ⊂ h L has polynomial decay of at least degree k , k x α ξ L k h L ≤ C, ξ ∈ S k ( F Ω ) , α ∈ N d , | α | ≤ k. Lemma 5.12.
Let χ be a smooth and real-valued bump-function such that supp( χ ) ⊂ B (0; r/ , χ ( x ) = χ ( − x ) , χ (0) = 1 and k χ k = 1 . (i) Extend χ to a function χ ∈ C c ( F Ω ) such that χ ( L , x ) = χ ( x ) . Then ( χ | χ ) G Del = 1 G Del . (ii) Given p ∈ R d , define the function χ p ∈ C c ( G Del , σ ) by χ p ( L , x ) = χ ( x − p ) . Then forany L ∈ Ω and p, q ∈ L , ω L ( χ p ∗ χ ∗ q ) = δ p,q .Proof. For part (i) we compute( χ | χ ) G Del ( L , y ) = Z R d χ ( L − z, − z ) χ ( L − z, y − z ) σ (( L , z ) , ( L − z, y − z )) d z. The integral will be zero unless − z, y − z ∈ L − z ∩ B (0; r/ y ∈ L , L ∈ Ω and L is uniformly r -discrete, this will only happen when y = 0. Then, because the 2-cocycle is 1when y = 0 and k χ k = 1, ( χ | χ ) G Del ( L , y ) = δ y, = 1 G Del ( L , y ).For part (ii) we compute that( χ p ∗ χ ∗ q )( L ,
0) = X y ∈L χ p ( L , y ) χ q ( L , y ) σ (( L , y ) , ( L − y, − y ))= X y ∈L χ ( y − p ) χ ( y − q ) = χ ( p − q ) = δ p,q where we have used that supp( χ ) ⊂ B (0 , r/
2) and L is r -discrete. (cid:3) iven ( L , y ) ∈ G Del , define the action ξ ( L ,y ) ( x ) = ξ ( L − x, y − x ) σ (( L − x, y − x ) , ( L − y, − y )) ∈ h L , ξ ∈ L ( F Ω ) . In the case that σ is comes from a magnetic twist that is constant over the unit space Ω Λ , wesee that ξ ( L ,y ) ( x ) = e − i h x,By i ξ ( L − x, y − x ) with B a real-valued skew-adjoint matrix describingthe magnetic field. Lemmas 4.3 and 5.12 now give the following. Lemma 5.13. (i)
Recall the functions { χ p } from Lemma 5.12. Then for any ξ ∈ L ( F Ω ) and ( L , p ) ∈ G Del , ξ ( L ,p ) = ( ξ · χ ∗ p ) L . (ii) Let e ∈ F L ( F Ω , σ ) G Del be such that ( e | e ) C ∗ r ( G Del ) = 1 C ∗ r ( G Del ) . Then for any L ∈ Ω the set { e ( L ,y ) } y ∈L is an orthonormal system in h L . Proposition 5.14.
Let ( e j ) j ∈ J be a (countable) right frame of F L ( F Ω , σ ) G Del . Then forall
L ∈ Ω , the set { e ( L ,y ) j } j ∈ J,y ∈L is a normalised tight frame for h L ∼ = L ( R d ) . If ( e j ) j ∈ J ⊂ A k S k ( F Ω , σ ) B k for k = 0 , · · · , ∞ , then the normalised tight frame { e ( L ,y ) j } has polynomial decayof at least order k .Proof. The first statement is a special case of Proposition 4.4. Lemma 5.11 ensures that theelements e ( L , j ( x ) = e j ( L − x, − x ) have polynomial decay of at least degree k . The translation e ( L ,y ) j ( x ) = e j ( L − x, y − x ) σ (( L − x, y − x ) , ( L − y, − y )) will have the same decay propertiesfor all y ∈ L . (cid:3) We can define an action π L : C (Ω Λ ) ⋊ R d on the localisation Hilbert space h L ∼ = L ( R d ) by( π L ( f ) ξ L )( x ) = (cid:0) f · ξ ) L ( x ) = ( f · ξ )( L − x, − x ) . Explicitly, we can compute that( f · ξ )( L − x, − x ) = Z R d f ( L − x, u − x ) ξ L ( u ) σ (( L − x, u − x ) , ( L − u, − u )) d u. Let us now consider the localisation π L ( p ) h ⊕ n L for p ∈ M n ( C ∗ r (Ω Λ ⋊ R d , σ )) a projection. Theorem 5.15.
Let k = 0 , · · · , ∞ and p = p ∗ = p ∈ M n ( A k ) . Then there are elements { e j } mj =1 ⊂ p S k ( F Ω , σ ) ⊕ n B k such that for all L ∈ Ω , { e ( L ,y )1 , . . . e ( L ,y ) m } y ∈L is a normalised tightframe of π L ( p ) L ( R d , C n ) with polynomial decay of at least degree k . The normalised tight frame { e ( L ,y )1 , . . . e ( L ,y ) m } y ∈L is an orthonormal basis for all L ∈ Ω if and only if p S k ( F Ω , σ ) ⊕ n B k ∼ = B ⊕ mk . For concreteness, we note that the case p ∈ M n ( A ) with A = A ∞ gives a normalised tight framewith faster than polynomial decay. Remark . Let us briefly consider the stabilityof Theorem 5.15 under deformations of the groupoid 2-cocycle σ using results from Gillaspy [14].Given the groupoid F = Ω Λ ⋊ R d , we can consider the trivial bundle of groupoids F × [0 , F × [0 ,
1] is a locally compact Hausdorff groupoid. A homotopy of groupoid 2-cocyclesis a groupoid 2-cocycle ω : ( F × [0 , (2) → T , which will give rise to a family of 2-cocycles { ω t } t ∈ [0 , on F that is continuous in t .Because Ω Λ ⋊ R d satisfies the Baum–Connes conjecture with coefficients, [14, Theorem 5.1]applies, which says that the evaluation map q t : C ∗ r ((Ω Λ ⋊ R d ) × [0 , , ω ) → C ∗ r (Ω Λ ⋊ R d , ω t )induces an isomorphism of K -theory groups. Composing this isomorphism with the Moritaequivalence of Ω Λ ⋊ R d with G Del , given a homotopy of 2-cocycles σ • on F × [0 ,
1] (and so on G Del × [0 ,
1] by restriction), we can consider finitely generated and projective modules P • over ∗ r ( G Del × [0 , , σ • ). Composing with the evaluation map, P is a free C ∗ r ( G Del , σ )-module ifand only if P is a free C ∗ r ( G Del , σ )-module.Considering the magnetic twists of Example 5.7, we can easily construct homotopies of 2-cocycles via a continuous map B • : Ω Λ × [0 , → V R d which restricts to a continuous path { B t } t ∈ [0 , of magnetic fields.5.3. The localisation dichotomy.
Decay properties of the normalised tight frame in Theorem5.15 come from the seminorms on the dense submodules S k ( F Ω , σ ) over the pre- C ∗ -algebras A k ⊂ C ∗ r (Ω Λ ⋊ R d , σ ) and B k ⊂ C ∗ r ( G Del , σ ) for k = 0 , · · · , ∞ . Because each A k is a pre- C ∗ -algebra, given p = p ∗ = p ∈ M n ( C ∗ r (Ω Λ ⋊ R d , σ )) and 0 < ε <
1, there is some p k = p ∗ k = p k ∈ M n ( A k ) ⊂ M n ( C ∗ r (Ω Λ ⋊ R d , σ )) such that k p − p k k < ε in C ∗ -norm as well as aunitary u k in M n ( C ∗ r (Ω Λ ⋊ R d , σ )) such that p k = u ∗ k pu k (see [7, Section 4]). Consequently thefinitely generated and projective C ∗ ( G Del , σ )-modules pL ( F Ω , σ ) ⊕ n G Del and p k L ( F Ω , σ ) ⊕ n G Del areisomorphic. The module p k L ( F Ω , σ ) ⊕ n G Del contains the dense submodule p k S k ( F Ω , σ ) ⊕ n B k . Notethat we can choose p k = p ∞ for all k ≥ C ∗ -module frames for S ∞ ( F Ω , σ ) ⊂ S ( F Ω , σ ) ⊂ L ( F Ω , σ ), we can prove aweak version of the localisation dichotomy considered in [24, Section 5 (arXiv version)] andfurther developed in [25]. Proposition 5.17 (Weak localisation dichotomy) . Let p = p ∗ = p ∈ M n ( C ∗ r (Ω Λ ⋊ R d , σ )) and p k = p ∗ k = p k ∈ M n ( A k ) be equivalent projections as above. The following statements areequivalent. (i) There is a C ∗ -module isomorphism pL ( F Ω , σ ) ⊕ n G Del ∼ = C ∗ r ( G Del , σ ) ⊕ m . (ii) There are elements { w j } mj =1 ⊂ p S ( F Ω , σ ) ⊕ n B such that for all L ∈ Ω , the collection { w ( L ,y )1 , . . . , w ( L ,y ) m } y ∈L is an orthonormal basis of π L ( p ) L ( R d , C n ) and for all y ∈ L , (8) m X j =1 Z R d (1 + | x − y | ) (cid:12)(cid:12) w ( L ,y ) j ( x ) (cid:12)(cid:12) d x < ∞ . (iii) There exists { e j } mj =1 ⊂ p ∞ S ∞ ( F Ω , σ ) ⊕ n B ∞ where for all L ∈ Ω , { e ( L ,y )1 , . . . , e ( L ,y ) m } y ∈L isan orthonormal basis of π L ( p ) L ( R d , C n ) with faster than polynomial decay.Proof. All statements except for Equation (8) immediately follow from Theorem 5.15. To seeEquation (8), we note that the frame elements are such that k w j · χ ∗ p k < ∞ for j ∈ { , . . . , m } , p ∈ R d and k · k the seminorm on S ( F Ω , σ ), k ξ k = k ξ k + d X l =1 k∇ l ξ k . Passing to the localisation, the functions w ( L ,y ) j and [ ∇ l ( w j · χ ∗ y )] L ( x ) = ( x l − y l ) w ( L ,y ) j ( x ) are in π L ( p ) L ( R d , C n ) for any y ∈ L , j ∈ { , . . . , m } and l ∈ { , . . . , d } . We can combine these casesto obtain Equation (8). (cid:3) Proposition 5.17 should be compared to the Localisation Dichotomy Conjecture in [24, Section5 (arXiv version)]. We have shown that an s ∗ -localised Wannier basis for s ∗ = 1 is equivalentto a Wannier basis with faster than polynomial decay, which in turn is equivalent to a freefinitely generated and projective module. To improve condition (iii) to an exponentially localisedWannier basis will require more analytic arguments that fall outside the framework of pre- C ∗ -algebras we have considered. In Section 5.5, we show that conditions (i)-(iii) of Proposition 5.17imply that the (even) noncommutative Chern numbers vanish. Remark . Theorem 5.15 and Proposition 5.17plays a similar role to the Balian–Low Theorem in time-frequency analysis. Brieflly, the theorem tates that if a Gabor system { e πimt g ( t − n ) } m,n ∈ Z forms an orthonormal basis of L ( R ), theneither g or the Fourier transform ˆ g is such that the sum in Equation (8) with m = 1 diverges. Bythe work of Luef [23], the Balian–Low Theorem can also be interpreted using finitely generatedand projective modules over C ∗ ( Z ) and the fact that C ( T ) has no non-trivial projections.See [23] for more details and a generalisation to the rotation algebra A θ ≃ C ∗ ( Z , θ ).5.4. Wannier bases for Hamiltonians on L ( R d , C n ) with Delone potentials. We modela particle in R d subject to a uniform magnetic field perpendicular to the sample. We take amagnetic potential A = ( A , . . . , A d ) such that A j ∈ L ( R d ) and differentiable with ∂∂x j A k − ∂∂x k A j = const.for all j, k ∈ { , . . . , d } . For simplicity, we consider constant magnetic field strength but moregeneral fields are possible (cf. Example 5.7). The magnetic Schr¨odinger operator is given by H = d X j =1 (cid:18) − i ∂∂x j − A j (cid:19) , We choose the symmetric gauge and define A j = − d P k =1 θ j,k x k for j = 1 , . . . , d , where θ j,k isantisymmetric and real. Our choice of gauge gives the magnetic translations { U a } a ∈ R d , wherefor any a ∈ R d , [ H , U a ] = 0 , ( U a ψ )( x ) = e − i h x,θa i ψ ( x − a ) , ψ ∈ L ( R d ) . Given a compact space Ω with action T : R d → Homeo(Ω), we can define the groupoid 2-cocycle θ : (Ω ⋊ R d ) (2) → T , θ (( ω, x ) , ( T − x ω, y )) = e − i h x,θy i , which is normalised, θ (( ω, x ) , ( T − x ω, − x )) = 1, and constant over the unit space.We wish to relate spectral properties of aperiodic Schr¨odinger operators to the Delone groupoid.We do this by considering atomic potentials on point sets,(9) H Γ = H + X p ∈ Γ v ( · − p ) , H = d X j =1 (cid:16) − i ∂∂x j − A j (cid:17) , where v an atomic potential function. Provided the potential V Γ = P p ∈ Γ v ( · − p ) is essentiallybounded, real valued and measurable, H Γ is essentially self-adjoint on the dense core C ∞ c ( R d ).We assume Γ is r -discrete and restrict our potentials to the K -subharmonic functions on R d , L K,r ( R d ) = n f ∈ L ( R d ) : | f ( x ) | ≤ Kr − d Z | x − y | L ∈ Ω .Recall the pre- C ∗ -algebra A = A ∞ ⊂ C ∗ r (Ω Λ ⋊ R d , θ ) that comes from the Fr´echet completionof C c (Ω Λ × R d , θ ) in the seminorms defined from the derivations { ∂ j } dj =1 (Proposition 2.10). Lemma 5.21. Let h be a self-adjoint element affiilated to M n ( C ∗ r (Ω Λ ⋊ R d , θ )) . Suppose that ∆ ⊂ σ ( h ) is a bounded spectral region separated from σ ( h ) \ ∆ with positive distance. Then p ∆ ( h ) = χ ∆ ( h ) ∈ M n ( A ) .Proof. Because ∆ is an isolated spectral region, p ∆ ( h ) can be approximated arbitrarily well by ϕ ( h ) with ϕ ∈ C ∞ c ( R ) such that ϕ ( x ) = 1 for x ∈ ∆ and ϕ ( x ) = 0 for x ∈ σ ( h ) \ ∆. Hence ϕ ( h ) ∈ M n ( A ). (cid:3) Hence, we can adapt Theorem 5.15 to the case of Schr¨odinger operators on L ( R d , C n ) withDelone atomic potentials. Theorem 5.22. Let Λ be a ( r, R ) -Delone set and let H Λ be a magnetic Schr¨odinger operator on L ( R d , C n ) with Delone atomic potential as in Equation (9) with v ∈ L K,r ( R d ) and uniformlycontinuous. Suppose that ∆ is an isolated and bounded spectral region of σ ( H L ) for all L ∈ Ω Λ .Then there are elements w , . . . , w m ∈ p ∆ S ∞ ( F Ω , θ ) ⊕ n such that for all L ∈ Ω the magnetictranslates { U y w L , . . . , U y w L m } y ∈L give a normalised tight frame of p ∆ ( H L ) L ( R d , C n ) with fasterthan polynomial decay.The frame { U y w L , . . . , U y w L m } y ∈L is an orthonormal basis of p ∆ ( H L ) L ( R d , C n ) for all L ∈ Ω if and only if the finitely generated and projective C ∗ -module p ∆ L ( F Ω , θ ) ⊕ n G Del ∼ = C ∗ r ( G Del , θ ) ⊕ m .Proof. By the spectral gap assumption, the family of spectral projections { p ∆ ( H L ) } L∈ Ω Λ givean element p ∆ ( h ) ∈ M n ( A ). As such, we can apply Theorem 5.15 which gives the fasterthan polynomially decaying tight frame or orthonormal basis of the localisation π L ( p ∆ ) h ⊕ n L = p ∆ ( H L ) L ( R d , C n ). (cid:3) Remarks . (i) The existence of an isomorphism p ∆ L ( F Ω , θ ) ⊕ m G Del ∼ = C ∗ r ( G Del , θ ) ⊕ m is a K -theoretic statement and implies that [ p ∆ ] = m [1] ∈ K ( C ∗ r ( G Del , θ )).(ii) If we take a deformation of the magnetic field { θ t } t ∈ [0 , such that the R -valued 2-cocycle ω t ( x, y ) = h x, θ t y i is continuous in t , we obtain a homotopy of 2-cocycles in the senseof Remark 5.16. Therefore, Theorem 5.22 is stable under deformations of the magneticfield provided that the spectral gap remains open throughout the deformation.5.5. Obstruction to localised Wannier basis by (noncommutative) Chern numbers. Let us briefly recall the periodic setting. If the atomic potential V Λ = P p ∈ Λ v ( · − p ) is suchthat Λ is a periodic and co-compact group G , then H Λ is affiliated to the algebra C ∗ r (Ω Λ ⋊ R d , θ ) ∼ = C ∗ r (( R d /G ) ⋊ R d , θ ) ∼ = C ∗ r ( G, θ ) ⊗ K . In the case G = Z d , the the non-triviality of finitely generated and projective C ∗ r ( Z d , θ )-moduleswith θ rational can be examined by studying the Chern classes of the spectral subspaces of theHamiltonian viewed as a complex vector bundle over the Brillouin torus, b Z d . In the aperiodicsetting, we can use tools from noncommutative geometry to carry out an analogous argu-ment. Indeed, noncommutative Chern numbers for Hamiltonians affiliated to C (Ω) ⋊ θ R d and C ∗ r ( G Del , θ ) have already been studied [11, 10, 9].Throughout this section, we will regularly take advantage of the equivalence between the contin-uous hull Ω Λ ⋊ R d and G Del , which gives an isomorphism K ( C ∗ r (Ω Λ ⋊ R d , θ )) ∼ = K ( C ∗ r ( G Del , θ )).We first recall the top degree noncommutative Chern numbers for aperiodic or disordered mag-netic Schr¨odinger Hamiltonians with a spectral gap. To do this, we recall the trace per unit olume on L ( R d ). Let Λ j be an increasing sequence of sets that converge to R d in an appro-priate sense, e.g. Λ j = [ − j, j ] d . Then for any a ∈ B ( L ( R d )),Tr Vol ( a ) = lim j →∞ j ) Tr(Π Λ j a ) , Π Λ j : L ( R d ) → L (Λ j ) . Proposition 5.24 ([11]) . Fix a probability measure P on Ω Λ that is invariant and ergodicunder the R d -action and let S d denote the permutation group of { , . . . , d } . If d > is even and p = p ∗ = p ∈ M n ( A ) , then for almost all L ∈ Ω Λ the functional (10) C d ( p ) = ( − πi ) d/ ( d/ X τ ∈ S d ( − τ (Tr C n ⊗ Tr Vol ) (cid:18) π L ( p ) d Y j =1 [ X τ ( j ) , π L ( p )] (cid:19) , is integer valued and almost surely constant in Ω Λ . The number C d almost surely defines a homomorphism K ( C ∗ r (Ω Λ ⋊ R d , θ )) → Z , which wecan also consider as a homomorphism K ( C ∗ r ( G Del , θ )) → Z . In particular C d ( p ) = C d ( p ′ ) if[ p ] = [ p ′ ] ∈ K ( C ∗ r ( G Del , θ )) and C d ( p ) = 0 if [ p ] = m [1] ∈ K ( C ∗ r ( G Del , θ )).For systems with d ≥ 3, we may also wish to consider lower-dimensional invariants. Theseinvariants are not integer-valued in general, but can still be used to study the topology ofgapped spectral projections. We fix a probability measure P on Ω Λ that is invariant and underthe R d -action. Then we recall the noncommutative calculus for A ⊂ C ∗ r (Ω Λ ⋊ R d , θ ), T ( f ) = Z Ω Λ f ( L , 0) d P , ( ∂ j f )( L , x ) = x j f ( L , x ) , f ∈ A , j ∈ { , . . . , d } . Proposition 5.25 ([11]) . Let p = p ∗ = p ∈ M n ( A ) and P a probability measure on Ω Λ that isinvariant and under the R d -action. Then for any k ≤ d an even integer, the functional (11) C k ( p ) = ( − πi ) k/ ( k/ X τ ∈ S k ( − τ (Tr C n ⊗T ) (cid:18) p k Y j =1 ∂ τ ( j ) p (cid:19) , defines a homomorphism K ( C ∗ r (Ω Λ ⋊ R d , θ )) → R . If P is ergodic under the R d -action and k = d , then C k ( p ) = C d ( p ) ∈ Z from Equation (10) almost surely. We again note that if [ p ] = m [1] ∈ K ( C ∗ r ( G Del , θ )), then C k ( p ) = 0 for any k ≥ 2. Combiningour results on the noncommuative Chern numbers with Theorem 5.15 and the weak localisationdichotomy (Proposition 5.17), we have the following. Corollary 5.26. Let p = p ∗ = p ∈ M n ( A ) and P a probability measure on Ω Λ that is invariantunder the R d -action. If C k ( p ) from Equation (11) is non-zero for some k ≥ , then for any L ∈ Ω there can not be Wannier basis of π L ( p ) L ( R d , C n ) constructed from magnetic translationsin L of elements { w j } mj =1 ⊂ p S ( F Ω , θ ) ⊕ n B such that for any y ∈ L m X j =1 Z R d (1 + | x − y | ) (cid:12)(cid:12) ( U y w L j )( x ) (cid:12)(cid:12) d x < ∞ . Deformation of the Delone atomic potential. We would like to consider the stabilityof our results on aperiodic Schr¨odinger operators H Λ when the Delone set Λ is deformed (e.g.from an aperiodic set to a periodic lattice). Deforming a Delone set Λ will change the crossedproduct groupoid and the K -theory may change substantially as the Delone set changes. How-ever, we will show the pairings in cyclic cohomology considered in Section 5.5 are unaffected bysuch deformations. Lemma 5.27. Let v ∈ C c ( R d ) be a continuous atomic potential with compact support. If { Λ t } t ∈ [0 , is continuous path of ( r, R ) -Delone sets in Del ( r,R ) , then the path of Schr¨odingeroperators { H Λ t } t ∈ [0 , is norm-continuous in the resolvent topology. roof. It is straightforward to see that for any v ∈ C c ( R d ), v ∈ L K,r ( R d ) and the function v Λ ( x ) = P p ∈ Λ v ( p − x ) is real-valued, measurable and essentially bounded for any ( r, R )-Deloneset Λ. Because Dom( H Λ ) is constant for any Λ ∈ Del ( r,R ) , we can use the resolvent identity tobound k ( z − H Λ ) − − ( z − H Λ ) − k = k ( z − H Λ ) − ( H Λ − H Λ )( z − H Λ ) − k≤ ess . sup | v Λ − v Λ | k ( z − H Λ ) − k k ( z − H Λ ) − k for any z ∈ C \ R . The result will therefore follow if we can show that the essential supremumis controlled by the topology on Del ( r,R ) . Suppose that supp( v ) ⊂ B (0; M ) for some M > ( r,R ) (Proposition 5.2) with d H the Hausdorff metric, we take x ∈ R d and suppose that d H (Λ − x ∩ B (0; M ) , Λ − x ∩ B (0; M )) ≤ δ < r/ 2. Taking δ small enough, we can ensure that | Λ − x ∩ B (0; M + r/ | = | Λ − x ∩ B (0; M + r/ | and,furthermore, we can decompose the sets Λ − x ∩ B (0; M + r/ 2) and Λ − x ∩ B (0; M + r/ 2) aspairs ( p, q ) ∈ Λ M := Λ ∩ B ( x ; M + r/ × Λ ∩ B ( x ; M + r/ 2) such that k p − q k ≤ δ . We cantherefore estimate | ( v Λ − v Λ )( x ) | = (cid:12)(cid:12)(cid:12) X p ∈ Λ v ( p − x ) − X q ∈ Λ v ( q − x ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) X p ∈ Λ ∩ B ( x ; M ) v ( p − x ) − X q ∈ Λ ∩ B ( x ; M ) v ( q − x ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) X ( p,q ) ∈ Λ M k p − q k≤ δ v ( p − x ) − v ( q − x ) (cid:12)(cid:12)(cid:12) ≤ X ( p,q ) ∈ Λ M k p − q k≤ δ (cid:12)(cid:12) v ( p − x ) − v ( q − x ) (cid:12)(cid:12) . By continuity of v , given any ε > δ so that (cid:12)(cid:12) v ( p − x ) − v ( q − x ) (cid:12)(cid:12) < ε | Λ ∩ B ( x ; M + r/ | for all k p − q k ≤ δ . Hence | ( v Λ − v Λ )( x ) | < ε and the essential supremum isalso bounded by ε . (cid:3) Definition 5.28. Let { Λ t } t ∈ [0 , be a continuous path in Del ( r,R ) . We say that { H Λ t } t ∈ [0 , is agapped path if there exists a bounded interval ∆ ⊂ R such that for all t ∈ [0 , 1] and L t ∈ Ω Λ t ,∆ ∩ σ ( H L t ) is non-empty and ∆ is separated from the rest of the spectrum of H L t by a positivedistance.The conditions to obtain a gapped path are quite strong, but if satisfied give a path of operators { h t } t ∈ [0 , such that h t is affiliated to C ∗ r (Ω Λ t ⋊ R d , θ ) and p t = χ ∆ ( h t ) ∈ A t ⊂ C ∗ r (Ω Λ t ⋊ R d , θ ),a dense pre- C ∗ -algebra. Proposition 5.29. Let { Λ t } t ∈ [0 , be a continuous path in Del ( r,R ) and fix an atomic potential v ∈ C c ( R d ) . Suppose that { H Λ t } t ∈ [0 , is a gapped path with isolated spectral region ∆ ⊂ R .Then for p t = { χ ∆ ( H L t ) } L t ∈ Ω Λ t ∈ A t and any even integer k ≤ d , the function [0 , ∋ t C k ( p t ) = ( − πi ) k/ ( k/ X τ ∈ S k ( − τ (Tr C n ⊗T ) (cid:16) p t k Y j =1 ∂ τ ( j ) p t (cid:17) ∈ R is continuous, where C k ( p ) is the weak Chern number from Equation (11) .Proof. The assumption on the spectral gap implies that C k ( p t ) is well-defined for all t . Becausewe have a uniform isolated spectral region ∆, we can write for all t ∈ [0 , p t = 12 πi I C ( z − h t ) − d z, h t = { ( z − H L t ) − } L t ∈ Ω Λ t with C a contour enclosing ∆ and not intersecting any other part of the spectrum. If { Λ t } t ∈ [0 , is a continuous path in Del ( r,R ) , then there is a corresponding continuous path {L t } t ∈ [0 , with L • in the orbit space of Λ • . Because t ( z − H L t ) − is norm-continuous by Lemma 5.27, so is t 7→ k ( z − h t ) − k . By the integral formula for the spectral projections, t 7→ k p t k is continuous.Because the functional C k induces a weaker topology than the norm topology, t C k ( p t ) isalso continuous. (cid:3) ontinuity of the function [0 , ∋ t 7→ k ( z − h t ) − k ∈ R for all z in the resolvent set impliesthat the spectral edges of σ ( h t ) are continuous away from gap closing points, see [3].Continuity of the Chern numbers under deformations of Delone sets means that if the range ofthe pairing is quantised, then it is constant under deformations. Corollary 5.30. Let { Λ t } t ∈ [0 , be a continuous path in Del ( r,R ) and fix an atomic potential v ∈ C c ( R d ) . Suppose that { H Λ t } t ∈ [0 , is a gapped path with isolated spectral region ∆ ⊂ R . Let P be an invariant and ergodic probability measure on Ω Λ . Then for almost all L ∈ Ω Λ , C d ( p t ) = C d ( p L ) := ( − πi ) d/ ( d/ X τ ∈ S d ( − τ (Tr C n ⊗ Tr Vol ) (cid:18) χ ∆ ( H L ) d Y j =1 [ X τ ( j ) , χ ∆ ( H L )] (cid:19) is integer valued and constant for all t ∈ [0 , . Corollaries 5.30 and 5.26 then give us the following stability result on delocalised Wannier bases. Corollary 5.31 (Stability of delocalised Wannier basis under atomic deformations) . Let Λ bean ( r, R ) -Delone set and consider H Λ with atomic potential v ∈ C c ( R d ) . Fix an invariant andergodic probability measure on Ω Λ and suppose that C d ( p L ) = 0 for almost all L ∈ Ω Λ . Thenfor any gapped path { H Λ t } t ∈ [0 , with isolated spectral region ∆ ⊂ R and any L t in the transversal Ω ,t , there can not be a faster than polynomially decaying Wannier basis of χ ∆ ( H L t ) L ( R d , C n ) built from magnetic translates in L t . We again note that by Proposition 5.17 the non-existence of a faster than polynomially decayingWannier basis also implies that the weaker localisation bound of Equation (8) diverges, m X j =1 Z R d (1 + | x | ) (cid:12)(cid:12) w L j ( x ) (cid:12)(cid:12) d x = ∞ . Acknowledgments. The authors thank Franz Luef, Domenico Monaco and Guo Chuan Thi-ang for valuable feedback on an earlier version of this manuscript. We also thank GiovannaMarcelli, Massimo Moscolari and Gianluca Panati for sharing the results of [24, 25] with us. CBis supported by a JSPS Grant-in-Aid for Early-Career Scientists (No. 19K14548) and thanks theMathematical Institute, Universiteit Leiden, for hospitality during the conference Noncommu-tative Geometry, Analysis, and Topological Insulators in February 2020, where this work tookshape. Both authors thank the Casa Matematica Oaxaca for hospitality and support duringthe workshop Topological Phases of Interacting Quantum Systems in June 2019. References [1] A. 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WPI-AIMR, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan and RIKEN iTHEMS,2-1 Hirosawa, Wako, Saitama 351-0198, Japan E-mail address : [email protected] Mathematisch Instituut, Universiteit Leiden, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands E-mail address : [email protected]@math.leidenuniv.nl