Gabor Duality Theory for Morita Equivalent $C^*$-algebras

The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalent $C^*$-algebras where the equivalence bimodule is a finitely generated projective Hilbert $C^*$-module. These Hilbert $C^*$-modules are equipped with some extra structure and are called Gabor bimodules. We formulate a duality principle for standard module frames for Gabor bimodules which reduces to the well-known Gabor duality principle for twisted group $C^*$-algebras of a lattice in phase space. We lift all these results to the matrix algebra level and in the description of the module frames associated to a matrix Gabor bimodule we introduce $(n,d)$-matrix frames, which generalize superframes and multi-window frames. Density theorems for $(n,d)$-matrix frames are established, which extend the ones for multi-window and super Gabor frames. Our approach is based on the localization of a Hilbert $C^*$-module with respect to a trace.


Introduction
Hilbert C * -modules are well-studied objects in the theory of operator algebras and Rieffel made the crucial observation that they provide the correct framework for the extension of Morita equivalence of rings to C * -algebras. In his seminal work [27] he noted that the equivalence bimodules between two C * -algebras are bimodules where the left and right Hilbert C * -module structures are compatible and the respective C * -valued inner products satisfy an associativity condition. We are interested in the features of these equivalence bimodules from the perspective of frame theory. In [12] the notion of standard module frame was introduced for countably generated Hilbert C * -modules. Rieffel has already in [28] observed that finitely generated equivalence bimodules may be described in terms of finite standard module frames and used it in his study of Heisenberg modules, which is a class of projective Hilbert C * -modules over twisted group C * -algebras. In [18] the properties of standard module frames for Heisenberg modules have been studied from the perspective of duality theory, which was motivated by the observation in [21] that these module frames are closely related to Gabor frames for an associated Hilbert space. Gabor frames have some additional features not shared by wavelets and shearlets that is due to the seminal contributions [9,19,29], where they developed the duality theory of Gabor frames.
Theorem (Duality Theorem). The Gabor system {e 2πiβlt g(t − αk)} k,l∈Z generated by a function g ∈ L 2 (R) is a frame for L 2 (R) if and only if {e 2πilt/α g(t − k/β)} k,l∈Z is a Riesz sequence for the closed span of {e 2πilt/α g(t − k/β)} k,l∈Z in L 2 (R) Here t ∈ R.
Due to its far-reaching implications there have been attempts to extend the duality principle to other classes of frames [1,15,3,4], see [6,8,31,32] for the theory of R-duals and [10].
Motivated by the link between the duality theory of Gabor frames and the Morita equivalence of noncommutative tori [21,18] we extend the duality theory of Gabor frames to module frames for equivalence bimodules between Morita equivalent C * -algebras. The setup for our duality theory has its roots in [18] and is as follows: Let A and B be C *algebras where B is assumed to have a unit and be equipped with a faithful finite trace tr B . We define a left Gabor bimodule to be a quadruple where E is a Morita equivalence A-B-bimodule. We show that module frames for Gabor bimodules admit a duality theorem and by localization with respect to a trace we are able to connect these module frame statements to results on frames in Hilbert spaces. Note that in [11] a different notion of localization of frames was introduced which constructs frames with additional regularity, which we also establish in our general setting. The main application of our duality results is a concise treatment of Gabor frames for closed cocompact subgroups of locally compact abelian phase spaces. Our general approach to duality principles has led us to the introduction of (n, d)-matrix Gabor frames that is a joint generalization of multi-window superframes and Riesz bases and we prove that their Gabor dual systems are (d, n)-matrix Gabor frames.
We say G(g; Λ) is an (n, d)-matrix Gabor frame for L 2 (G) if the collection of time-frequency shifts G(g; Λ) is a frame for L 2 (G×Z n ×Z d ). Equivalently, there exists h ∈ L 2 (G×Z n ×Z d ) such that for all f ∈ L 2 (G × Z n × Z d ) we have f r,k , π(λ)g l,k L 2 (G) π(λ)h l,s dλ, for all r ∈ Z n and s ∈ Z d . We develop the theory of these (n, d)-matrix Gabor frames and prove a duality theorem for this novel type of frames. Let us summarize the content of this paper. In Section 2 we collect some facts about C *algebras, Hilbert C * -modules, the localization of Hilbert C * -modules and finitely generated projective Hilbert C * -modules. In Section 3 we introduce Gabor bimodules, and study the case when these have a single generator in terms of module frames. We establish the analog of the duality theorem for Gabor frames for Gabor bimodules with one generator. In Section 3.2 we extend all of these results to the case of finitely many generators which leads one naturally to matrix-valued extensions of the statements and definitions in the preceding section. We also prove a density theorem for module frames. In the final section, Section 4, we discuss applications to Gabor frames for closed subgroups of the time-frequency plane of locally compact abelian groups.
2. Preliminaries on C * -algebras and Hilbert C * -modules We assume basic knowledge about Banach * -algebras, C * -algebras, and of Banach modules and Hilbert C * -modules. In this section we collect definitions and basic facts of concepts crucial for this paper, such as positivity in C * -algebras, Morita equivalence of C * -algebras, and localization of Hilbert C * -modules. For these topics we refer to [20], [25], and [24].
For a C * -algebra A and a ∈ A, we denote by σ A (a) the spectrum of a in A. We will need the following important result.
where A ·, · is the A-valued inner product. Also, whenever c ≥ 0 in A, we have a * ca ≤ c a * a for all a ∈ A.
Proposition 2.7 ( [25], Corollary 2.22). Let A be a C * -algebra. If E is a Hilbert A-module and T ∈ End A (E), then for any f ∈ E as elements of the C * -algebra A, where A ·, · is the A-valued inner product.
Suppose φ is a positive linear functional on a C * -algebra B, and that E is a right Hilbert B-module. We define an inner product where ·, · B is the B-valued inner product. We may have to factor out the subspace and complete E/N φ with respect to ·, · φ . This yields a Hilbert space which we will denote by H E . This is known as the localization of E in φ. There is a natural map ρ φ : E → H E which induces a map ρ φ : End B (E) → B(H E ). We will focus entirely on the case in which φ is a faithful positive linear functional, that is, when b ∈ B + and φ(b) = 0 implies b = 0. In that case N φ = {0} and we have the following useful result from [20, p. 57-58].
Proposition 2.8. Let A be a C * -algebra equipped with a faithful positive linear functional φ : A → C, and let E be a left Hilbert A-module. Then the map ρ φ : The Hilbert C * -modules of interest will be of a very particular form in that they will be A-B-equivalence bimodules for C * -algebras A and B. We will denote the A-valued inner product by • ·, · , and the B-valued inner product by ·, · • . Definition 2.9. Let A and B be C * -algebras. A Morita equivalence bimodule between A and B, or an A-B-equivalence bimodule, is a Hilbert C * -module E satisfying the following conditions.
Now let A ⊂ A and B ⊂ B be dense Banach * -subalgebras such that the enveloping C * -algebra of A is A, and the enveloping C * -algebra of B is B. Suppose further there is a dense A-B-inner product submodule E ⊂ E such that the conditions above hold with A, B, E instead of A, B, E. In that case we say E is an A-B-pre-equivalence bimodule.
We will make repeated use of the following fact in the sequel without mention. Proposition 2.10 ( [25], Proposition 3.11). Let A and B be C * -algebras and let E be an It is a well-known result that if E is an A-B-equivalence bimodule, then B ∼ = K A (E) through the identification Θ f,g → f, g • . Here Θ f,g is the compact module operator Θ f,g : h → • h, f g. We make particular note of the case when E is a finitely generated Hilbert A-module.
Proposition 2.11. Let E be an A-B-equivalence bimodule. Then E is a finitely generated projective A-module if and only if B is unital.
Proof. Suppose first A is finitely generated and projective as a Hilbert A-module. As E is finitely generated, any A-endomorphism on E is determined by its action on a finite set of generators. Hence End A (E) = K A (E), and the former is unital.
Conversely, we assume that B is unital. As B ∼ = K A (E), and the latter is an ideal in for all h ∈ E. It follows that E is a finitely generated projective A-module.
Note that the systems {f 1 , ..., f n } and {g 1 , ..., g n } are not necessarily A-linearly independent, but they still provide a reconstruction formula: z = n i=1 • h, f i g i . Motivated by spanning sets in finite-dimensional vector spaces, also called frames, we call the system {f 1 , ..., f n } a module frame for E and {g 1 , ..., g n } is referred to as a dual module frame. The properties of module frames for equivalence bimodules are the main objective of this work.
The following two results concern properties of module frames consisting of a single element, though we do not formally introduce module frames until Definition 3.8. For our setup it will turn out that it is enough to consider module frames consisting of only one element, see Section 3. The results will come into play when we relate module frames and Gabor frames in Section 4. Lemma 2.12. Let A be any C * -algebra, and let E be a (left) Hilbert A-module. Suppose T ∈ End A (E) is such that there exist C, D > 0 such that for all f ∈ E. Then T is invertible, and Proof. By (2.1), we see that T is positive and invertible with C Id E ≤ T ≤ D Id E . Positivity is preserved when multiplying by positive commuting operators, so it follows that 13. Let A be any C * -algebra, and let E be a (left) Hilbert A-module. Let T ∈ End A (E) be such that there exist C, D > 0 such that for all f ∈ E. Then the smallest possible value of D is T , and the largest possible value for C is It follows that the smallest value for D is T . By Lemma 2.12 we see by the same argument that the minimal value for 1 C is T −1 . Hence the largest value for C is T −1 −1 . It is an interesting question when the property of being finitely generated projective passes to dense subalgebras and corresponding dense submodules.
Proposition 2.14. Let E be an A-B-equivalence bimodule as in Definition 2.9, with B unital. Suppose there are dense Banach * -subalgebras A ⊂ A and B ⊂ B, where B is spectral invariant in B and has the same unit as B. Suppose further that E ⊂ E is an A-B-pre-equivalence bimodule. If E is a finitely generated projective A-module, then E is a finitely generated projective A-module.
Proof. The latter part of the proof of Proposition 2.11 can be adapted to this situation. The full proof can be found in [28,Proposition 3.7].
Since we aim to mimic the situation of Gabor analysis, which we will treat in Section 4, the positive linear functional which we localize our Morita equivalence bimodule with respect to will have a particular form. In particular it will be a faithful trace. For unital Morita equivalent C * -algebras A and B Rieffel showed in [26] that there is a bijection between non-normalized finite traces on A and non-normalized finite traces on B under which to a trace tr B on B there is an associated trace tr A on A satisfying Here E is the Morita equivalence bimodule. We will in the sequel almost always consider A or B unital, and so instead we will suppose the existence of a finite faithful trace on one C * -algebra (the unital one) and induce a possibly unbounded trace on the other C * -algebra. The following was proved in [2, Proposition 2.7] and ensures that this procedure works.
Proposition 2.15. Let E be an A-B-equivalence bimodule, and suppose tr B is a faithful finite trace on B. Then the following hold: i) There is a unique lower semi-continuous trace on A, denoted tr A , for which for all f, g ∈ E. Moreover, tr A is faithful, and densely defined since it is finite on span{ • f, g : f, g ∈ E}. Setting for f, g ∈ E defines inner products on E, with f, g tr A = f, g tr B for all f, g ∈ E. Consequently, the Hilbert space obtained by completing E in the norm f ′ = tr A ( • f, f ) 1/2 is just the localization of E with respect to tr B .
If both C * -algebras are unital then the induced trace is also a finite trace as in [26], see [2, p. 8].
Convention 2.16. We have the following as a standing assumption for the rest of the manuscript unless otherwise specified: Suppose we have a faithful trace tr B on a unital C * -algebra B, and an A-B-equivalence bimodule E. If A is a unital C * -algebra, we pick the normalization of tr B such that tr A becomes a state. That is, we normalize the trace on the C * -algebra on the left in the Morita equivalence.

Gabor bimodules
3.1. The single generator case. Throughout this section we discuss properties of equivalence bimodules of the following type.
Definition 3.1. Let A and B be C * -algebras where B is assumed to have a unit and is equipped with a faithful finite trace tr B . We define a left Gabor bimodule to be a quadruple where E is an A-B-equivalence bimodule. We define a right Gabor bimodule analogously, that is, it is a quadruple where A is unital, tr A is a faithful finite trace on A, and E is an A-B-equivalence bimodule. Indeed, this will be of great importance in the sequel, and we will use tr A without mentioning that it is induced by tr B . Likewise with tr B if we treat the case of right Gabor bimodules.
We are also interested in Gabor bimodules possessing some regularity, see Section 4. We define a right Gabor bimodule with regularity analogously.
The rest of this section will be devoted to exploring properties of Gabor bimodules, mostly left Gabor bimodules. In Section 4 we show that Gabor bimodules over twisted group C * -algebras for LCA groups are Rieffel's Heisenberg modules and provide a different approach to Gabor analysis. We start with some basic definitions from [12]. We restrict to a single generator in this section, and extend the results to finitely many generators in Section 3.2. Indeed, we will see in that section that even the case of finitely many generators can be reduced to the case of a single generator of an associated Morita equivalence bimodule.
Definition 3.4. For g ∈ E we define the analysis operator by (3.4) and the synthesis operator : An elementary computation shows that Φ * g = Ψ g .
Remark 3.5. As E is an A-B-bimodule, we could just as well have defined the analysis operator and the synthesis operator with respect to the B-valued inner product. Indeed we will need this later, but it will then be indicated by writing Φ B g . Unless otherwise indicated the analysis operator and synthesis operator will be with respect to the left inner product module structure.
Definition 3.6. For g, h ∈ E we define the frame-like operator Θ g,h to be The frame operator of g is the operator Remark 3.7. The module frame operator Θ g is a positive operator since Θ g = Φ * g Φ g .
Definition 3.8. We say g ∈ E generates a (single) module frame for E if Θ g is an invertible operator E → E. Equivalently, there exist constants C, D > 0 such that holds for all f ∈ E.
Remark 3.9. When {g} is a module frame for E, Θ g is a positive invertible operator on E.
What follows will largely be a study of g and h in E such that Θ g,h is invertible, and how this relates to inner product inequalities for a localization H E of E. Our interest in this question is due to the fact that in certain cases module frames can be localized to obtain Hilbert space frames, see Section 4. We begin with a result which generalizes the Wexler-Raz biorthogonality condition for Gabor frames, which we also look at in Section 4. Proof. Suppose f = Θ h,g f = Θ g,h f for all f ∈ E. Then E is a finitely generated projective A-module as it is generated by g, and we can use g and h to make the maps r and s from the proof of Proposition 2.11. Hence B ∼ = K A (E) = End A (E), and as End A (E) is unital, we deduce B is unital. By Morita equivalence which finishes the proof.
The following result showcases a duality between certain A-submodules of E and Bsubmodules of E, which is very particular to our setting of Morita equivalence bimodules.
Proposition 3.11. For any g, h ∈ E the following two statements are equivalent.
Proof. Suppose first • f, h g = f for all f ∈ Ag. By Morita equivalence of A and B, f = f h, g • for all f ∈ Ag, hence h, g • fixes all elements in Ag. In particular, since A has an approximate unit, g ∈ Ag, so g h, g • = g. Now let f ′ ∈ gB. We then write f ′ = gb for some b ∈ B, and so we deduce since g h, g • = g by the above. We extend the reconstruction formula to all of gB by continuity. The proof of the converse is completely analogous.
In the special case where Proposition 3.11 (i) holds for all f ∈ E we get another reconstruction formula. Note the (subtle) difference in the placement of g and h in the statement compared to statement (ii) in the preceding proposition.
Then E is finitely generated and projective as an A-module as before, since it is singly generated by g, and we may use g and h to make the maps r and s from the proof of Proposition 2.11. Hence B ∼ = K A (E) = End A (E) and B is unital. We may rewrite the equality to f = f h, g • for all f ∈ E, which implies h, g • = 1 B as B acts faithfully on E. But then Then if we let f ∈ hB we may write f = hb for some b ∈ B, and we get We extend the reconstruction formula to hB by continuity.
Note that in the setting of Proposition 3.11 we may of course interchange g and h. However the subspaces Ah and Ag do not need to coincide. We may however guarantee Ag = Ah when h has a special form.
Proof. Let f ∈ Ag. As Θ g , and hence also Θ g | Ag , is an A-module operator, so is Θ g | −1 Ag . Thus we get Hence we have Ag ⊂ Ah. Also g ∈ Ag as A has a left approximate unit, and as Θ g is invertible as a map Ag → Ag it follows that h = Θ g | −1 Ag g ∈ Ag. Hence Ag = Ah.
In the remaining part of the article we focus mostly on the case where Θ g,h is invertible as a map E → E.
Given g ∈ E such that Θ g : E → E is invertible and with h = Θ −1 g g the canonical dual atom, one may ask what the canonical dual atom of h is. The following lemma tells us that it is exactly what one would expect from Hilbert space frame theory.
Proof. With h = Θ −1 g g the identity Θ h g = h is established as follows: We proceed to show that Θ h is invertible. By Lemma 3.13 Ag = Ah, and we know Ag = E. Now let f ∈ Ag, and write f = ag for some a ∈ A. We then have Ag is dense in E, and by extending the reconstruction formulas to all of E by continuity it follows that Θ −1 h = Θ g . Then the canonical dual atom of h is which proves the result.
The following proposition tells us that g and Θ −1 g g then indeed have the desired properties as described in Proposition 3.11.
Proof. Note first that E = Ag = Ah by Lemma 3.13. Now, let f ∈ Ah and write f = ah for some a ∈ A. Then we have which holds by Lemma 3.16. We extend the reconstruction formula by continuity so it is valid for all f ∈ E. By Proposition 3.11 this also implies (3) is also true.
We may also prove the following additional reconstruction formula when h is the canonical dual atom. Note the (subtle) difference in where h and g are in the reconstruction formula compared to Proposition 3.11. Proposition 3.18. Let g ∈ E be such that Θ g is invertible, and let h = Θ −1 g g. Then f = h g, f • for all f ∈ gB and f ′ = g h, f ′ • for all f ′ ∈ hB. As a consequence, gB = hB.
Proof. Suppose f in gB. For g and h as stated, we have f = • f, h g for all f ∈ E. By Proposition 3.11 we then have f = g h, f • for all f ∈ gB. Then The second statement follows from noting that our assumptions imply Ag = Ah = E by Lemma 3.13 and the fact that the canonical dual of h is g by Lemma 3.16. Then we may simply interchange g and h in the argument for the first assertion. Lastly we prove gB = hB. We know g ∈ gB as B has an approximate unit, so Likewise, h ∈ hB and so This finishes the proof.
There is a correspondence between projections in Morita equivalent C * -algebras, see for example [28]. We formulate the following variant. Let E be an A-B-equivalence bimodule, and let B be unital. Then there is a way of constructing idempotents in A. This is the content of the following proposition.
One of the cornerstones of Gabor analysis is the duality principle, see for example [9,19,29]. One of the main intentions of this investigation is a reformulation of this duality principle in our module framework. To this end we introduce the following operator. For an element g ∈ E we define the B-coefficient operator by Note that this operator is B-adjointable with adjoint We are now in the position to state and prove the module version of the duality principle. ( Proof. We show that both statements are equivalent to g, g • being invertible in B. Suppose Θ g is invertible. Then E is finitely generated and projective as an A-module, as we can make the maps r and s from the proof of Proposition 2.11 using g and Θ −1 g g. Thus B is unital. As On the other hand, (2) implies that B is unital and the statement is equivalent to g, g • being invertible in B.
In Gabor analysis one is often concerned with the regularity of the atoms generating a Gabor frame, see Section 4. In case g is so that Θ g is invertible on all of E with g ∈ E, and B ⊂ B is spectral invariant Banach * -subalgebra with the same unit as B, the canonical dual atom has the following important property.
We deduce that g, g • is invertible in B and But as g ∈ E we have g, g • ∈ B. By spectral invariance of B in B it follows that g, g • −1 ∈ B. Then, since EB ⊂ E, it follows that which is the desired assertion.
3.2. Extending to several generators. We extend the above theory to several generators. Indeed we will lift the A-B-equivalence bimodule E to an M n (A)-M d (B)-equivalence bimodule, for d, n ∈ N, and consider a type of module frame in this matrix setting. We will see in Section 4 that this generalizes n-multiwindow d-super Gabor frames of [18]. Note that we will index an n × d-matrix by (i, j), i ∈ Z n , j ∈ Z d , that is, we start indexing at 0. The reason for this is that in Section 4 we will need to incorporate the groups Z k , k ∈ N. Here Z k denotes the group Z/(kZ). We The action of M n (A) on M n,d (E) is defined in the natural way, that is for a ∈ M n (A) and f ∈ M n,d (E). Likewise we define an M d (B)-valued inner product on M n,d (E) in the following way Indeed it is not hard to verify the three conditions of Definition 2.9. Verifying conditions ii) and iii) is a matter of verifying the statements in each matrix element using that E is an A-Bequivalence bimodule. Verifying condition i) is a matter of getting density in each matrix entry by choosing elements of M n,d (E) in the correct way. Namely, if we want to get the elements in place (i, j) in M n (A), then we may for example pair elements of M n,d (E) with nonzero entry only in place (i, k) with elements of M n,d (E) with nonzero entry only in place (j, k), for some k ∈ Z d . The analogous procedure holds for M d (B). Density then follows by • E, E = A and E, E • = B. In particular, we have for f, g, h ∈ M n,d (E) that and also Also, since the new inner products are defined using the inner products • −, − and −, − • , we see that in case we have Banach * -subalgebras A ⊂ A and B ⊂ B, as well as an A-B-subbimodule E ⊂ E as above, we get as well as We wish to reduce the matrix algebra case to the Gabor bimodule case of Section 3.1, so we need to guarantee that spectral invariance of Banach * -subalgebras lifts to matrices. For convenience we include the following result. For g ∈ M n,d (E) we define as in Section 3.1 the analysis operator which has as adjoint the operator Using these we also define the frame-like operator As noted in Section 3.1, Θ g is a positive operator. For simplicity, and since it is the case we will most often consider, suppose in the following that B is unital with a faithful finite trace. There is then an induced (possibly unbounded) trace on A as in Section 2. We may lift these traces to the matrix algebras. Indeed, there are traces on M n (A) and M d (B) satisfying (3.22) tr for all f, g ∈ M n,d (E). They are given by (3.23) The trace on M d (B) extends to a finite trace on the whole algebra, but the same might not be true for the densely defined trace on M n (A). It is however true if A, and hence also M n (A), is unital.
is also a faithful trace.
We summarize the preceding discussion in the following proposition which allows us to study the M n (A)-M d (B)-equivalence bimodule M n,d (E) by studying the A-B-bimodule E. with the above defined actions, inner products, and traces is also a left Gabor bimodule with regularity. The analogous statements hold for right Gabor bimodules.
Since our focus is on the description of frames in equivalence bimodules for Morita equivalent C * -algebras, we want to do this now on the matrix algebra level and thus introduce an appropriate notion of module frames for the matrix-valued equivalence bimodules.
Definition 3.26. Let g = (g i,j ) i∈Zn,j∈Z d ∈ M n,d (E). We say g generates a module (n, d)matrix frame for E with respect to A if there exists h = (h i,j ) i∈Zn,j∈Z d ∈ M n,d (E) for which (3.25) f r,s = k∈Z d l∈Zn holds for all f = (f i,j ) i∈Zn,j∈Z d ∈ M n,d (E), r ∈ Z n , and s ∈ Z d . Even though all results of Section 3.1 lift to the induced matrix algebra setup, we want to discuss explicitly two results relating the lifted traces. We show in Section 4 that these two results extend the density theorems of Gabor analysis to Gabor bimodules. Since we in Theorem 3.28 talk about left Gabor bimodules and in Theorem 3.29 talk about right Gabor bimodules, we will for the sake of avoiding confusion not have any convention on the normalization of traces in the two results. Proof. The assumption that Θ g is invertible implies [g, g ] • is invertible. Then  Proof. The assumptions imply • [ g, g] −1 ∈ M n (A), so it follows as in Section 3.1 that , , as well as (3.23), we get 3.3. From a Gabor bimodule to its localization. In [21] the existence of multi-window Gabor frames for L 2 (R d ) with windows in Feichtinger's algebra was proved through considerations on a related Hilbert C * -module. Furthermore, in [22] projections in noncommutative tori were constructed from Gabor frames with sufficiently regular windows. Thus being able to pass from an equivalence bimodule E to a localization H E and back is quite important, and we dedicate this section to results on this procedure. We will interpret this in terms of standard Gabor analysis in Section 4, and we will explain how L 2 (G), for G a second countable LCA group, relates to H E for specific modules E which arise in the study of twisted group C * -algebras. We denote by (−, −) E the inner product on the localization of E in tr A . Concretely, we have (f, g) E = tr A ( • f, g ).
In other words, g is a module frame for E if and only if the inequalities in (3.30) are satisfied for some C, D > 0.
Proof. Suppose first that there is an h ∈ E such that • f, g h = f for all f ∈ E. By Morita equivalence this implies for all f ∈ E. As before, this implies 1 B = g, h • = h, g • . Since tr B is a positive linear functional we obtain for all f ∈ E, where we have used Proposition 2.7 to deduce We then get the lower frame bound with C = • h, h −1 , that is for all f ∈ E. By Proposition 2.8 all intermediate steps involve operators that extend to bounded operators on H E , so we may extend by continuity. We get the upper frame bound by use of Proposition 2.7 in the following manner Conversely, suppose there are C, D > 0 such that The assumption implies that f → f g, g • is a positive, invertible operator on H E . By Proposition 2.1 it follows that g, g • is invertible in B. Thus f → f g, g • is a positive, invertible operator on E as well. Hence the operator Define h := Θ −1 g g, and let f ∈ E be arbitrary. Then we have from which the result follows.
We are interested in module frames and module Riesz sequences, and their relationship to frames and Riesz sequences in Gabor analysis for LCA groups. To get results on Riesz sequences in Section 4 we need a module version of Riesz sequences which, when localized, yields the Riesz sequences we know from Gabor analysis. For this we let A be unital with a faithful trace tr A , and we need to localize A as a Hilbert A-module in the trace tr A . We let (a 1 , a 2 ) A := tr A (a 1 a * 2 ). The completion of A in this inner product will be denoted H A , and the action of A on H A is the continuous extension of the multiplication action (from the right) of A on itself.  Proof. First suppose Φ g Φ * g : A → A is an isomorphism. Then, since by Proposition 2.6 • ag, ag = a • g, g a * ≤ • g, g aa * , we may deduce Hence in (3.31) we may set D = • g, g . Since Φ g Φ * g : A → A is an isomorphism and Φ g Φ * g a = a • g, g , it follows that there is • g, g −1 ∈ A. Then (a, a) A = tr A (aa * ) A = tr A (a • g, g 1/2 • g, g −1 • g, g 1/2 a * ) ≤ • g, g −1 tr A (a • g, g a * ) = • g, g −1 tr A ( • ag, ag ) = • g, g −1 (ag, ag) E , which implies that we may set C = • g, g −1 −1 in (3.31). All intermediate steps extend to H A by Proposition 2.8.
Suppose now that (3.31) is satisfied. The lower inequality in (3.31) tells us that for all a ∈ A, Note that we need the upper inequality of (3.31) to extend all intermediate steps to H A via Proposition 2.8. It follows that • g, g is a positive invertible operator on H A ⊃ A. By Proposition 2.1 it follows that • g, g is invertible in A. Then, since Φ g Φ * g a = a • g, g , it follows that Φ g Φ * g : A → A is an isomorphism.
Both Proposition 3.30 and Proposition 3.31 were proved for Gabor bimodules, so by Remark 3.27 the results lift to the corresponding matrix setting of Section 3.2.
Remark 3.32. Note that in the proofs of the two preceding results the upper bounds in (3.30) and (3.31) were both satisfied with D = • g, g . We will see in Section 4 that in the Gabor analysis setting, this means that all atoms coming from the Hilbert C * -module are Bessel vectors for the localized frame system. For use in Section 4, we introduce the following notion.
Definition 3.34. Let (A, B, E, tr A ) be a right Gabor bimodule, and let g ∈ E. If Φ g Φ * g : A → A is an isomorphism, we say g generates a module Riesz sequence for E with respect to A. If h ∈ M n,d (E) generates a module Riesz sequence for M n,d (E) with respect to M n (A), we will also say that h generates a module (n, d)-matrix Riesz sequence for E with respect to A.

Applications to Gabor analysis
In this section we show how the above results reproduce some of the core results of Gabor analysis for LCA groups. We will see how some of the cornerstones of Gabor analysis on LCA groups are trivial consequences of the above framework. Of particular interest is the reproduction of some of the main results of [18] on n-multiwindow d-super Gabor frames with windows in the Feichtinger algebra. Indeed we will show the corresponding results for localized module (n, d)-matrix frames, which generalize n-multiwindow d-super Gabor frames.
To present the results we will need to explain how time frequency analysis on LCA groups relates to Morita equivalence of twisted group C * -algebras. In the interest of brevity, we refer the reader to [18] for a more in-depth treatment of time frequency analysis and its relation to twisted group C * -algebras, and to [16] for a survey on the Feichtinger algebra.
Throughout this section, we fix a second countable LCA group G and let G be its dual group. We fix a Haar measure µ G on G and normalize the Haar measure µ G on G such that the Plancherel theorem holds. By Λ we denote a closed subgroup of the time-frequency plane G × G. The induced topologies and group multiplications on Λ and (G × G)/Λ turn them into LCA groups as well, and we may equip them with their respective Haar measures. Having fixed the Haar measures on G, G, and Λ, we will assume (G × G)/Λ is equipped with the unique Haar measure such that Weil's formula holds, that is, such that for all f ∈ L 1 (G × G) we have In this setting we can define the size of Λ by

Note that s(Λ) is finite if and only if Λ is cocompact in G × G.
For any x ∈ G and ω ∈ G we define the translation operator (or time shift) T x by and the modulation operator (or frequency shift) E ω by These operators are unitary on L 2 (G), and satisfy the commutation relation For any ξ = (x, ω) ∈ G × G we may then define the time-frequency shift operator We define the 2-cocycle where we have introduced the symplectic cocycle c s by We also remark that (4.7) π(ξ) * = c(ξ, ξ)π(−ξ) for all ξ ∈ G × G.
Using the symplectic cocycle c s we define for a closed subgroup Λ ⊂ G × G the adjoint subgroup Λ • by (4.8) Then (Λ • ) • = Λ and Λ • ∼ = (G × G)/Λ, see for example [17]. Note that Λ is cocompact if and only if Λ • is discrete. With these identifications we put on Λ • the Haar measure such that the Plancherel theorem holds with respect to Λ • and (G × G)/Λ. We define the short time Fourier transform with respect to g ∈ L 2 (G) as the operator for ξ ∈ G × G. The Feichtinger algebra S 0 (G) is then defined by A norm on S 0 (G) is given by It is a nontrivial fact that all elements of S 0 (G)\{0} determine equivalent norms on S 0 (G).
In case G is discrete it is known that S 0 (G) = ℓ 1 (G) with equivalent norms. Furthermore, S 0 (G) consists of continuous functions and is dense in both L 1 (G) and L 2 (G).
For two functions F 1 , F 2 over Λ ⊂ G × G we define the twisted convolution by (4.12) and the twisted involution F * 1 (λ) := c(λ, λ)F 1 (−λ). In [18] it was shown that S 0 (Λ) is a Banach * -D-algebra for some D > 0 when equipped with twisted convolution, and indeed it is possible to choose an equivalent norm on S 0 (G) such that it becomes a Banach *algebra. We denote the resulting Banach * -algebra by S 0 (Λ, c). Using this we may then define two Banach * -algebras Note that A ∼ = S 0 (Λ, c) and B ∼ = S 0 (Λ • , c) via the natural maps. We will use these identifications without mention in the sequel.
The following was proved in [18].
is a faithful unitary c-projective representation of Λ. As a result, the integrated representation is a non-degenerate * -representation of S 0 (Λ, c).
We may then obtain the minimal universal enveloping algebra C * r (Λ, c) of S 0 (Λ, c) through the integrated representation of S 0 (Λ, c) on L 2 (G), that is, the representation for a ∈ S 0 (Λ, c) and f ∈ L 2 (G). As Λ is abelian, hence amenable, the minimal and maximal enveloping algebras coincide, so we write C * (Λ, c) for the universal enveloping algebra of S 0 (Λ, c). We do the same for S 0 (Λ • , c), and denote its universal enveloping C * -algebra by C * (Λ • , c). Indeed, S 0 (G) becomes a pre-equivalence bimodule between A and B as in Definition 2.9 when equipped with the inner products and the actions with a ∈ A, b ∈ B, and f, g ∈ S 0 (G). That these are well-defined was noted in Section 3 of [18]. In the remainder of the section we denote by A the C * -completion of A, B the C * -completion of B, E = S 0 (G), and by E the Hilbert C * -module completion of E. Hilbert C * -modules E as in this setting are called Heisenberg modules.
Remark 4.3. The fact that we get the same twisted group C * -algebras by using S 0 (Λ, c) as we get when using the more traditional approach with L 1 (Λ, c) was noted in [2].
Since S 0 -functions are continuous, there are also well-defined canonical faithful traces on A and B given by (4.19) and In general, these traces do not extend to A and B, but we will nonetheless denote them by tr A and tr B . These are indeed related as in (2.3), which can be seen from (4.21) tr In our discussion the following two results are crucial. The first follows immediately by [23], and the second is a consequence of [14].
Remark 4.6. Although the traces tr A and tr B do not in general extend to the algebras A and B, we can guarantee they extend in one case. Namely, tr A extends to all of A if A is unital, which is equivalent to Λ being discrete. The same is of course true for B and tr B , with the discreteness condition on Λ • . This is due to the fact that the trace given by evaluation in the identity extends to twisted group C * -algebras when the underlying group is discrete [5, p. 951].
The case of Λ or Λ • being discrete is the case we will almost exclusively restrict to after Proposition 4.8.
The following is now an immediate consequence. Proposition 4.7. Let Λ be cocompact, which implies Λ • is discrete. Then under the above conditions on A, B, E, tr B the quadruple (A, B, E, tr B ) is a left Gabor bimodule. In addition, the septuple (A, B, E, tr B , A, B, E) is a left Gabor bimodule with regularity.
If Λ • is cocompact and thus Λ is discrete, then we obtain a right Gabor bimodule with regularity analogously.
We may then reprove Theorem 3.9 of [18] in this framework.
Proposition 4.8. Let Λ ⊂ G × G be a closed subgroup. Then E is a finitely generated projective A-module if and only if Λ ⊂ G × G is a cocompact subgroup. Also, E is a finitely generated projective A-module if and only if Λ ⊂ G × G is cocompact.
Proof. E is finitely generated and projective over A if and only if K A (E) = End A (E). As E is an A-B-equivalence bimodule, this is equivalent to B being unital by Proposition 2.11. B is unital if and only if Λ • is discrete by Lemma 4.4, so equivalently Now, if Λ is cocompact, then B is unital, so we are in the situation of Proposition 2.14 by Proposition 4.5. Hence by the first part of this proposition it follows that E is a finitely generated projective A-module.
Conversely, suppose E is a finitely generated projective A-module. Then E ∼ = A n p isometrically for some n ∈ N and some p ∈ M n (A). Passing to the completions we obtain E ∼ = A n p, so E is a finitely generated projective A-module. By the first part of this proposition it follows that Λ is cocompact.
Remark 4.9. Proposition 4.8 shows that we can only have finite module frames for E as an A-module if Λ is cocompact in G × G. Since we wish to study the relationship between finite module frames and Gabor frames this is the case we care most about in the sequel.
To get results on Gabor frames for L 2 (G) with windows in E from the above setup, we will need to localize certain subsets of the C * -algebras A and B, as well as the Morita equivalence bimodule E, just as explained in Section 2. For simplicity, let Λ be cocompact in G × G from now on, unless otherwise specified. Then Λ • is discrete and tr B is defined on all of B. The localization of B in tr B is induced by the inner product (−, −) B given by Since B is dense in B and tr B is continuous, it follows that their localizations in tr B are the same. For b 1 , b 2 ∈ B we then have , we may identify the localization H B of B with ℓ 2 (Λ • ). By [2, Proposition 3.2] we also obtain that the localization of E in tr B is L 2 (G). Note that this is the same as the localization of E in tr A by construction, and that there is an action of A on L 2 (G) by extending the action of A on E.
It is slightly more tricky to localize subsets of A. Indeed, it is not in general possible as the trace might not be defined everywhere. However, even if A is not unital we may localize the algebraic ideal • E, E ⊂ A in the trace tr A . Indeed, by [2, Theorem 3.5], elements of E are such that whenever g ∈ E and f ∈ L 2 (G), then { f, π(λ)g } λ∈Λ ∈ L 2 (Λ). This is the property of being a Bessel vector, which we will discuss in more detail below. Hence for any f, g ∈ E, we may identify • f, g ∈ A with ( f, π(λ)g ) λ∈Λ in L 2 (Λ) by doing the analogous procedure with tr A as for tr B above.
We may do the same for the matrix algebras and matrix modules considered in Section 3.2. Note that • [ M n,d (E), M n,d (E)] = M n,d ( • E, E ). Adapting the setting of twisted group C * -algebras and Heisenberg modules above to the matrix algebra setting of Section 3.2 we see that we obtain the following identifications We can finally treat the analogs of Gabor frames in our framework. In what follows we will consider a novel type of Gabor frames. To ease notation we will for f ∈ L 2 (G × Z n × Z d ) write f i,j instead of f (·, i, j), and the same for elements of L 2 (Λ × Z n × Z n ) and (4.24) and the synthesis operator D g by m∈Zn Λ a k,m (λ)π(λ)g m,l dλ} k∈Zn,l∈Z d . Furthermore, we define the frame-like operator S g,h = D h C g , and for brevity we write S g for D g C g . We say S g is the frame operator associated to g.
We say g generates an (n, d)-matrix Gabor frame for L 2 (G) with respect to Λ if S g : is an isomorphism. Equivalently, the collection of time-frequency shifts (4.26) G(g; Λ) := {π(λ)g i,j | λ ∈ Λ} i∈Zn,j∈Z d is a frame for L 2 (G×Z n ×Z d ). We then say that G(g; Λ) is an (n, d)-matrix Gabor frame for f r,k , π(λ)g l,k π(λ)h l,s dλ, for all r ∈ Z n and s ∈ Z d . When g and h satisfy (4.27) we say G(g; Λ) and G(h; Λ) are a dual pair of (n, d)-matrix Gabor frames. If Λ is implicit, we may also say h is a dual (n, d)-matrix Gabor atom for g, or just a dual atom of g. Remark 4.13. When G(g; Λ) is an (n, d)-matrix Gabor frame for L 2 (G), there is always a dual (n, d)-matrix Gabor atom for g, namely h = S −1 g g. This is known as the canonical dual of g.
Remark 4.14. One can verify that C g = D * g . Thus S g is always a positive operator between Hilbert spaces, just as for the module frame operator in Section 3.
For general g ∈ L 2 (G × Z n × Z d ) the operator C g will not be bounded. Elements g such that C g is bounded are of interest on their own. Definition 4.15. If g ∈ L 2 (G×Z n ×Z d ) is so that C g : L 2 (G×Z n ×Z d ) → L 2 (Λ×Z n ×Z n ) is a bounded operator we say g is an (n, d)-matrix Gabor Bessel vector for L 2 (G) with respect to Λ, or that G(g; Λ) is an (n, d)-matrix Gabor Bessel system for L 2 (G). Equivalently, there is D > 0 such that for all f ∈ L 2 (G × Z n × Z d ) we have (4.28) f, f ≤ D C g f, C g f , which may also be written as The smallest D > 0 such that the condition of (4.28) holds is called the optimal Bessel bound of G(g; Λ), or just the optimal Bessel bound of g if the set Λ is clear from the context.
The Gabor frames of Definition 4.11 seemingly generalize the n-multiwindow d-super Gabor frames of [18]. Indeed, we obtain n-multiwindow d-super Gabor frames if we only require reconstruction of f ∈ L 2 (G × Z d ) and we identify L 2 (G × Z d ) ⊂ L 2 (G × Z n × Z d ) by embedding along a single element of Z n . Hence (4.27) generalizes both multiwindow Gabor frames and super Gabor frames as well, setting d = 1 or n = 1, respectively. However, we will in Proposition 4.34 show that any n-multiwindow d-super Gabor frame for L 2 (G) with respect to Λ is an (n, d)-matrix Gabor frame for L 2 (G) with respect to Λ. However, we continue to call them by separate names, since, as mentioned above, they are used for reconstruction in different Hilbert spaces.
The following proposition was noted in the (n, 1)-matrix case in [2,Theorem 3.11], and its proof in the (n, d)-matrix Gabor case goes through the same except with more bookkeeping.
Proposition 4.16. Let Λ ⊂ G × G be closed and cocompact. For every g ∈ M n,d (E), is a bounded operator. In other words, every g ∈ M n,d (E) is a Bessel vector.
For ease of notation, the localization map in in M n (A) will be denoted by ρ Mn(A) , even though we might not be able to localize all of M n (A). With the above definitions, the following calculation is justified for f, g ∈ M n,d (E) ⊂ L 2 (G × Z n × Z d ) by Proposition 4.16 Hence we obtain the following result.
Lemma 4.17. Let Λ ⊂ G × G be closed and cocompact. For every g ∈ M n,d (E), the module coefficient operator Φ g localizes to give the coefficient operator C g . Equivalently, the diagram Cg commutes for all g ∈ M n,d (E).

Likewise one may obtain
Note that the domain might be larger, but we cannot guarantee this unless A is unital, that is, when Λ is discrete.
Lemma 4.18. Let Λ ⊂ G × G be closed and cocompact. For every g ∈ M n,d (E), the module synthesis operator Φ * g localizes to the Gabor synthesis operator C * g . Equivalently, the diagram commutes for all g ∈ M n,d (E).
) is a linear bijection intertwining both the A-actions and the B-actions, we see by Proposition 4.19 that for g ∈ M n,d (E), Θ g is invertible if and only if S g | ρ M n,d (E) (M n,d (E)) is invertible. But we also have the following result. is invertible if and only if Since any g ∈ M n,d (E) is a Bessel vector by Proposition 4.16, we may extend the operator by continuity to obtain that S g : Conversely, suppose S g : Since S g is the continuous extension of Θ g , it then follows by Proposition 2.8 and Proposition 2.1 that Θ g is invertible, which implies S g | ρ M n,d (E) (M n,d (E)) is invertible.
Remark 4.21. From now on we will identify M n,d (E) and its image in the localization, and we will do this without mention.
Combining Proposition 4.19 and Lemma 4.20 we obtain the following important result.
Proposition 4.22. Let Λ ⊂ G × G be closed and cocompact. For g ∈ M n,d (E) we have that Θ g is invertible if and only if S g is invertible. In other words, g generates a module (n, d)-matrix frame for E with respect to A if and only G(g; Λ) is an (n, d)-matrix Gabor frame for L 2 (G).
We also have the following important corollary. Proof. Suppose first g, h ∈ M n,d (E) generate dual (n, d)-matrix Gabor frames for L 2 (G) with respect to Λ. Then we know that for all f ∈ M n,d (E) we have from which we as before deduce that [g, h ] • = 1 M d (B) . This extends by continuity to the identity operator on all of L 2 (G × Z n × Z d ).
Conversely, if [g, h ] • extends to the identity operator on L 2 (G × Z n × Z d ), then [g, h ] • acts as the identity on M n,d (E). For any f ∈ M n,d (E) we then have hence (4.27) holds for all f ∈ M n,d (E). But this extends to L 2 (G × Z n × Z d ) by continuity, which implies that g and h generate dual (n, d)-matrix Gabor frames.
We wish to establish a duality principle for (n, d)-matrix Gabor frames. For this we also need to treat (n, d)-matrix Gabor Riesz sequences and relate them to Definition 3.34.
Definition 4.24. Let g ∈ L 2 (G × Z n × Z d ). We say g generates an (n, d)-matrix Gabor Riesz sequence for L 2 (G) with respect to Λ, or that G(g; Λ) is an (n, d)-matrix Gabor Riesz sequence for L 2 (G), if a r,s (µ) = i∈Z d j∈Zn Λ a r,j (λ)π(λ)g j,i dλ, π(µ)h s,i for all r, s ∈ Z n and all µ ∈ Λ. If (4.31) is satisfied we will say h generates a dual (n, d)-matrix Gabor Riesz sequence of g.
Before treating localization of module matrix Riesz sequences and how they relate to matrix Gabor Riesz sequences, we do a necessary but justified simplification. Recall that existence of finite module matrix Riesz sequences for M n,d (E) with respect to M n (A) requires A to be unital by Proposition 3.20. In the following we therefore let Λ be discrete, but not necessarily cocompact. Hence in the following, A is unital with a faithful trace, but B might not have that property. By [17, p. 251] we know that G(g; Λ) is a Bessel system with Bessel bound D if and only if G(g; Λ • ) is a Bessel system with Bessel bound D. Applying Proposition 4.16 we immediately get the following from Lemma 4.17 and Lemma 4.18. As ) is invertible. Since any g ∈ M n,d (E) is a Bessel vector by Proposition 4.16, we may extend the operator by continuity to obtain that C g C * g : Conversely, suppose C g C * g : L 2 (Λ×Z n ×Z n ) → L 2 (Λ×Z n ×Z n ) is invertible. Since C g C * g is the continuous extension of Φ g Φ * g , it then follows by Proposition 2.8 and Proposition 2.1 that Φ g Φ * g is invertible as well, which implies (C g C * g )| ρ   Let Λ ⊂ G × G be discrete. For g ∈ M n,d (E) we have that Φ g Φ * g is invertible if and only if C g C * g is invertible. In other words, g generates a module (n, d)matrix Riesz sequence for E with respect to A if and only if G(g; Λ) is an (n, d)-matrix Gabor Riesz sequence for L 2 (G).
By the proof of Lemma 4.28 we then have the following statement.
Corollary 4.31. Let Λ ⊂ G × G be discrete. Suppose g, h ∈ M n,d (E). Then g and h generate dual (n, d)-matrix Gabor Riesz sequences for L 2 (G) with respect to Λ if and only if • [ g, h] extends to the identity operator on L 2 (Λ × Z n × Z n ).
Proof. Suppose first that g and h generate dual (n, d)-matrix Gabor Riesz sequences for L 2 (G) with respect to Λ. Then for all a ∈ M n (A) we have (a r,s ) = { i∈Z d j∈Zn Λ a r,j (λ)π(λ)g j,i dλ, π(µ)h s,i } µ∈Λ,r,s∈Zn , which is equivalent to a = a • [ g, h] for all a ∈ M n (A). But the first expression extends by continuity to L 2 (Λ × Z n × Z n ), so • [ g, h] extends to the identity on L 2 (Λ × Z n × Z n ).
Conversely, suppose • [ g, h] extends to the identity on L 2 (Λ × Z n × Z n ). Once again, for all a ∈ M n (A) we then have (a r,s ) = { i∈Z d j∈Zn Λ a r,j (λ)π(λ)g j,i dλ, π(µ)h s,i } µ∈Λ,r,s∈Zn , which again extends to L 2 (G × Z n × Z n ). Hence g and h are dual (n, d)-matrix Gabor Riesz sequences for L 2 (G) with respect to Λ.
Note how the above results guarantee that when Λ ⊂ G× G is closed and cocompact and g ∈ M n,d (E) is such that G(g; Λ) is an (n, d)-matrix Gabor frame for L 2 (G), the canonical dual frame S −1 g g ∈ M n,d (E). Indeed, Likewise, for Riesz sequences there is the notion of canonical biorthogonal atom, see for example [7, p. 160]. Restricting to Λ discrete, it is given by (S B g ) −1 g, where S B g is the frame operator with respect to the right hand side, that is, with respect to Λ • . We see that for all f ∈ M n,d (E)

Thus it follows that
. Hence for both matrix Gabor frames and matrix Gabor Riesz sequences with generating atom in M n,d (E), the canonically associated dual atoms are also in M n,d (E).
We have the following result which shows that in the cases we are interested in, if the generating atom is regular, the canonical dual atom has the same regularity. i) If G(g; Λ) is an (n, d)-matrix Gabor frame for L 2 (G) and Λ is closed and cocompact in G × G, then the canonical dual atom is in M n,d (E). ii) If G(g; Λ) is an (n, d)-matrix Gabor Riesz sequence for L 2 (G) and Λ is discrete, then the canonical biorthogonal atom is also in M n,d (E).
Proof. For the proof of i), note that the assumption that Λ is cocompact implies that Λ • is discrete, so by Lemma 4.4 we get that M d (B) is unital. Also M d (B) is a C * -subalgebra of B(H M n,d (E) ) by Proposition 2.8. That G(g; Λ) is an (n, d)-matrix Gabor frame for L 2 (G) then means that (3.30) is satisfied for our current setting. We deduce, as in the proof of Proposition 3.30, that [g, g ] • is invertible in M d (B). Since For the proof of ii), note that the assumption that Λ is discrete implies M n (A) is unital. Also, M n (A) is a C * -subalgebra of B(H Mn(A) ) by Proposition 2.8. That G(g; Λ) determines an (n, d)-matrix Gabor Riesz sequence for L 2 (G) then means that (3.31) is satisfied for our current setting. The middle term of (3.31) can be written (a • [ g, g], a) A , so • [ g, g] extends to a positive, invertible operator on H Mn(A) . We deduce as in the proof of Proposition 3.31 that • [ g, g] is invertible in M n (A). Since g ∈ M n,d (E) we have • [ g, g] ∈ M n (A), and by Remark 4.33. In the special case n = d = 1 Proposition 4.32 gives a proof of the fact that the canonical dual atom of a Gabor frame vector in Feichtinger's algebra S 0 (G) is also in Feichtinger's algebra whenever Λ is cocompact.
When applying the module setup of Section 3 to Gabor analysis, we take as a preequivalence bimodule E = S 0 (G × Z n × Z d ), which is a proper subspace of L 2 (G × Z n × Z d ) unless G is a finite group. Even the Hilbert C * -module completion E is properly contained in L 2 (G × Z n × Z d ) for most choices of Λ, see [2, Example 3.8]. As such, we cannot hope to treat general atoms in L 2 (G × Z n × Z d ) by applying just this method. But indeed the module reformulation is made exactly to guarantee some regularity of the atoms generating frames.
From Definition 4.11 we see that (n, d)-matrix Gabor frames generalize n-multiwindow d-super Gabor frames considered in [18]. However, we now make clear how they fit into the module framework. As mentioned earlier, we obtain n-multiwindow d-super Gabor frames if we only require reconstruction of f ∈ L 2 (G × Z d ) and we identify L 2 (G × Z d ) ⊂ L 2 (G × Z n × Z d ) by embedding it along a single element in Z n . The module reformulation of this is that g, h ∈ M n,d (E) are dual n-multiwindow d-super Gabor frames if for all f ∈ M n,d (E) supported only one row we have Likewise, it is clear that the (n, d)-matrix Gabor Riesz sequences of Definition 4.24 generalize the n-multiwindow d-super Gabor Riesz sequences also considered in [18]. Indeed, we obtain n-multiwindow d-super Gabor Riesz sequences if we only require reconstruction of a ∈ L 2 (Λ × Z n ) and we identify L 2 (Λ × Z n ) ⊂ L 2 (Λ × Z n × Z n ) by embedding it along a single element in the middle copy of Z n . The module reformulation of this is that g, h ∈ M n,d (E) are dual n-multiwindow d-super Gabor Riesz sequences if for all a ∈ M n (A) supported only one row we have We proceed to prove that all n-multiwindow d-super Gabor frames for L 2 (G) with respect to Λ are (n, d)-matrix Gabor frames for L 2 (G) with respect to Λ, as well as the analogous statement for Riesz sequences. The converse statement is true as well. i) If G(g; Λ) is an n-multiwindow d-super Gabor frame for L 2 (G) with a dual window h ∈ M n,d (E), then G(g; Λ) is an (n, d)-matrix Gabor frame for L 2 (G) with dual window h. ii) If G(g; Λ) is an n-multiwindow d-super Gabor Riesz sequence for L 2 (G) with a dual Gabor Riesz sequence G(h; Λ) with h ∈ M n,d (E), then G(g; Λ) is an (n, d)-matrix Gabor Riesz sequence for L 2 (G) with dual Gabor Riesz sequence G(h; Λ).
Proof. If G(g; Λ) is an n-multiwindow d-super Gabor frame for L 2 (G) with respect to Λ with a dual window h ∈ M n,d (E), we can, as noted above, reconstruct any f ∈ M n,d (E) supported on a single row. In other words, for all f ∈ M n,d (E) supported on a single row. Given arbitrary f ′ ∈ M n,d (E) we may then just write f ′ as a sum of n matrices f ′ i , i = 0, . . . , n − 1, with only one nonzero row, namely the kth row of f ′ i is given by (f i,0 , . . . , f i,d−1 )δ ik for k ∈ Z n , and we can reconstruct each of these rows. Hence we can reconstruct arbitrary elements of M n,d (E). This passes to the localization L 2 (G × Z n × Z d ), and thus finishes the proof of (i). The proof of (ii) is completely analogous, writing elements a ∈ M n (A) as a sum of matrices with only one nonzero row and then using that we can reconstruct such matrices. This will also pass to the localization L 2 (Λ × Z n × Z n ).
Given a closed and cocompact subgroup Λ, we may ask if there are restrictions on n, d ∈ N for there to possibly exist (n, d)-matrix Gabor frames for L 2 (G) with respect to Λ. Conversely, if we fix n and d, we may ask if there are restrictions on the size of the subgroup Λ for there to possibly exist (n, d)-matrix Gabor frames for L 2 (G) with respect to Λ. When Λ is a lattice, we have the following proposition. Proposition 4.35. Let Λ ⊂ G × G be a lattice. If there is g ∈ M n,d (E) such that G(g; Λ) is an (n, d)-matrix Gabor frame for L 2 (G), then s(Λ) ≤ n d .
Proof. Since Λ is discrete and cocompact, both A and B are unital. We also know by Proposition 4.32 that the canonical dual of g is in M n,d (E). Hence we are in the setting of Theorem 3.28. Since module (n, d)-matrix frames localize to (n, d)-matrix Gabor frames for the localization, and we have tr A (1 A ) = 1, and tr B (1 B ) = s(Λ) (since the identity on B is s(Λ)δ 0 , where δ 0 is the indicator function in the group identity, see for example [28]), the result is immediate by Theorem 3.28.
Likewise, given a lattice Λ, we may ask if there is a relationship between the size of Λ and the integers n and d such that there can possibly exist (n, d)-matrix Gabor Riesz sequences for L 2 (G) with respect to Λ. This is the content of the following proposition.
Proposition 4.36. Let Λ ⊂ G × G be a lattice. If g ∈ M n,d (E) is such that G(g; Λ) is an (n, d)-matrix Gabor Riesz sequence for L 2 (G), then s(Λ) ≥ n d .