Modulation Spaces as a Smooth Structure in Noncommutative Geometry

We demonstrate that a class of modulation spaces are examples of a smooth structure on the noncommutative 2-torus in the sense of recent developments in KK-theory. In addition, we prove that this class of modulation spaces can be represented as corners in operator linking algebras.


Introduction
The interplay between Gabor analysis and noncommutative geometry [Con94] has been explored earlier and the interplay is a rich one, see for example [Lue09], [DLL15], [Lue15]. Indeed, problems in Gabor analysis can often effectively be rephrased to operator algebraic questions. Moreover, Gabor analysis provides a way to generate projective modules over noncommutative tori [Lue09]. As so, Gabor analysis provides interesting examples of structures studied in operator algebra theory and noncommutative geometry. The main part of this paper focuses on the latter. Indeed, we show that Gabor analysis provides a way of generating C k -modules studied as part of recent research in unbounded KKtheory [Kaa14], [Kaa16], [Mes14], [BKM18]. In terms of Gabor analysis the notion of C k -modules over noncommutative tori translates into better localization of the window function generating the frame. It is common to refer to a Gabor frame generated by a Gaussian as better than one generated by a triangle function. Our results turns this observation into a rigorous statement Moreover, we show that modulation spaces are operator * -correspondences in the sense of [BKM18], which seems to be the first example of an operator * -correspondence not based on Riemannian manifolds. We then use a recently proven representation theorem to realize certain modulation spaces as subspaces of bounded operators on a Hilbert space, up to a suitable notion of isomorphism. Modulation spaces have turned out to be a convenient setting for time-frequency analysis and thus our results indicate that modulation spaces might be as well a natural class of function spaces in noncommutative geometry. Modulation spaces have recently been interpreted in terms of square-integrable representations of Hilbert C * -modules [Hua18].
The article is structured as follows. Section 2 and Section 3 are dedicated to introducing most of the relevant notions from noncommutative geometry, for example operator * -algebras, operator * -modules, and the desired notion of smoothness. In Section 4 a treatment of the noncommuative 2-torus, its structure, and its relation to time-frequency analysis is given. Here we also obtain its desired smooth structure. We explore modulation spaces in Section 5, as well as a brief treatment of relevant notions from Gabor analysis. In particular, we demonstrate that modulation spaces are C k -modules over the noncommutative 2-torus, the first main result of the article. Lastly, in Section 6 we explore another structure from noncommutative geometry on the modulation spaces and obtain a representation theorem for modulation spaces as corners in operator linking algebras.

Operator * -Algebras
In applications, we are often forced to deal with algebras which are pre-C * -algebras. We may then always pass to the C * -completion by going to the universal C * -envelope. When working with C * -algebras, there is a unique norm on a * -algebra making it a C * -algebra. By the Gelfand-Naimark theorem, any C * -algebra A can be realized as a norm-closed *subalgebra of B(H), the bounded operators on a Hilbert space H. This then gives a unique norm on the matrix algebras M n (A), since it can be realized as a closed * -subalgebra of M n (B(H)) ∼ = B(H n ). Passing to the C * -envelope we usually get a more well-behaved object to work with, but we might forget crucial underlying geometrical information present in the pre-C * -algebra. As an example, the C * -envelope of both C ∞ (T) and C 1 (T) is C(T), the continuous functions on the unit circle. But the pre-C * -algebras C ∞ (T) and C 1 (T) are very different. To remedy this, we consider another completion on the pre-C * -algebras which preserves more of this geometric data, namely we pass to an operator * -algebra. It will be clear from the definitions below that C * -algebras are operator * -algebras. First we need the notion of an operator space. A good reference on operator spaces, operator algebras and operator modules is [BLM04].
A linear map φ : X → Y between two operator spaces X and Y is called completely bounded when there exists a constant C > 0 such that For such a map, we define We say such a φ is completely isometric if every amplification φ n : M n (X) → M n (Y ) is an isometry. We say φ is a complete isomorphism if each φ n is a bounded isomorphism with bounded inverse.
Axioms ii) and iii) of Definition 2.1 are often called Ruan's axioms. Ruan showed that any Banach space X equipped with a family of norms {||.|| X,n }, one for each matrix dimension (and with ||.|| X,1 equal to the Banach space norm), is completely isometrically isomorphic to a closed subspace V ⊂ B(H) for some Hilbert space H [Rua88].
We wish to consider algebra structures, and so we adopt the following definition.
Definition 2.2. An operator algebra is an operator space A equipped with a multiplication map A × A → A such that i) A becomes a Banach algebra over C ; ii) We have the inequality for all m ∈ N, and all x, y ∈ M m (A), where we have defined multiplication in the matrix algebras by the formula x ik · y kj for all i, j ∈ {1, . . . m}. Note that this is equivalent to requiring that the algebra M m (A) is a Banach algebra over C for all m ∈ N.
We might also write xy instead of x · y from now on. The operator algebras we consider are pre-C * -algebras. In particular, they will be equipped with an involution, and so we adopt the following definition from [Kaa16].
3. An operator algebra A is an operator *-algebra when it comes equipped with an involution * : Blecher showed in [Ble95] that any operator algebra is completely isometrically isomorphic (as operator algebras) to a concrete operator algebra, that is, a closed subalgebra of B(H) for some Hilbert space H. Recently it was also shown [BKM18] that any operator * -algebra is completely isomorphic to a concrete operator *-algebra. By this we mean there is a completely bounded algebra isomorphism φ from the operator * -algebra A onto some closed subalgebra of B(H), and a selfadjoint unitary U ∈ B(H), such that U φ(a * )U = φ(a) * for all a ∈ A.
We shall need to consider modules over operator * -algebras. They mimic Hilbert C *modules in that they are equipped with an inner product. Unlike Hilbert C * -modules however, the norm on the module is in general not determined by the inner product.
Definition 2.4. Let A and B be operator algebras. An operator space X is an operator The module actions are compatible with the operator space norm in the sense that where the module actions of matrix algebras on matrix modules is defined in the natural way: In particular, a left operator A-module is an operator A − C-bimodule.
If A is an operator * -algebra, we say a left operator A-module X is a left operator * -module if it is equipped with a sesquilinear pairing A ·, · : X × X → A satisfying the conditions iii) A a · x, y = a · A x, y for all x, y ∈ X and all a ∈ A; iv) A x · λ + y · µ, z = A x, z · λ + A x, z · µ for all x, y, z ∈ X and all λ, µ ∈ C; v) A x, y = A y, x * vi) The inner product is compatible with the matrix norm structures via a generalized Cauchy-Schwarz inequality: for all m ∈ N and all x, y ∈ M m (X), where we have defined the matrix valued inner products as Lastly, two operator * -modules X and Y over A are cb-isomorphic if there exists a completely bounded bijective A-linear map φ : X → Y with a completely bounded inverse φ −1 , such that The above definitions can be adapted to right modules in the natural way. Indeed we will need this in Section 6.

Smoothness in Noncommutative Geometry
Much of recent research in KK-theory has been focused on obtaining an algebraic formula for the interior product in unbounded KK-theory [BJ83]. Mesland found in his thesis [Mes14] such an expression as long as the C * -algebras are restricted to a certain "smooth" subclass. Later, some more progress in this direction has been achieved for example in [Kaa14], [Kaa16], and some recent focus has been on a notion of smoothness in noncommutative geometry. We therefore dedicate this section to defining the suitable notion of smoothness suited for our purposes.
Definition 3.1. A C k -structure on a C * -algebra A is an inverse system of operator algebras where the structure maps are spectral invariant completely bounded * -homomorphisms with dense range. Now given a C * -algebra A, fix a spectral triple (H, D) for A, where H is a Hilbert space and D : H → H is a densely defined selfadjoint operator. There is then a natural C k -structure on A obtained by considering  Definition 3.2. A C k -algebra is a C * -algebra equipped with a fixed C k -spectral triple. We refer to this as the natural C k -structure. We say the C * -algebra is smooth if it is C k for all k ∈ N.
The above A k 's may be equipped with an operator * -algebra structure such that Definition 3.1 is satisfied. Indeed, the link to operator algebras is one of the strengths of Mesland's notion of smoothness. Explicitly describing this structure requires quite a bit of work. Since we will not have explicit use for for this operator * -algebra structure, we refer the interested reader to Section 4.1 of [Mes14].
Along with a notion of C k -algebras there is a notion of C k -modules, which reflect the C k -structure on the C * -algebra. This is nothing but a smoothness requirement on an approximate unit of A.
Definition 3.3. Fix a smooth C * -algebra A, where the smooth structure is given by {A i } i∈N . We say a Hilbert A-module E is a C k -A-module if there exists an approximate unit , and for which there exists constants C k such that Here we have defined x i ⊗ x i as the "rank-one operator" y → A y, x i x i . We say E is a smooth C * -module if there is an approximate unit that makes it a C k -module for all k ∈ N.
Note that for both modules and algebras, if it is C k , then it is also C m for m ≤ k. This definition allows us to obtain a chain of submodules of E Here H A j is the set of sequences (a i ) i∈N , a i ∈ A j for all i, for which where e k,l is the matrix with 1 in position (k, l) and zero elsewhere. This will be our notion of smooth submodules of a Hilbert C * -module, and the norm we put on E k is exactly the induced H A k -norm. It is known [Mes14] that the inclusions E k → E k−1 are completely contractive, and that E k becomes an operator module over A k when A k is defined from a spectral triple and given the operator * -algebra structure mentioned above.

The Noncommutative
Torus and Its C k -Subalgebras 4.1. The Noncommutative 2-Torus. The noncommutative 2-torus A θ is the universal C * -algebra generated by two unitaries u and v, satisfying the commutation relation for θ ∈ R. Denote by S(Z 2 ) the rapidly decaying sequences indexed by Z 2 , that is The * -algebra is then a Fréchet pre-C * -algebra when equipped with the seminorms We call this the smooth noncommutative 2-torus. We also want to place special emphasis on * -subalgebras of A θ which we will denote by A s θ , s ≥ 0. These are defined by (13) On both A ∞ θ and A s θ , s ≥ 0, there is a faithful trace τ given by This trace also satisfies τ (a * a) > 0 for 0 = a ∈ A ∞ θ , and τ (1) = 1. As so, it is a tracial state, and it gives a faithful, cyclic GNS-representation by defining H ′ := A ∞ θ as the vector space, and taking the completion with respect to the norm induced by the inner product This completion is denoted by H τ in the sequel. Note that since A ∞ θ is dense in A s θ for all s ≥ 0 for the C * -algebra norm, and by the following norm estimate, we obtain the same Hilbert space H τ by taking the completion of A s θ by the norm induced by τ .
The smooth noncommutative 2-torus A ∞ θ is also contained in the domain of the two canonical derivations on A θ . We denote these by δ 1 and δ 2 . They are defined by Note that τ • δ i = 0, and δ i (a * ) = δ i (a) * for all a ∈ Dom(δ i ), i = 1, 2. Also A ∞ θ is exactly the smooth domain for the derivations δ 1 and δ 2 . By definition of the derivations the following is immediate.
2 ) for all s ≥ max(n 1 , n 2 ). We will briefly discuss a particular spectral triple on the noncommutative 2-torus. Details can be found in [Va06] and [CPR11].
Consider H τ ⊕ H τ equipped with a Z 2 -grading given by the grading operator and let A θ act diagonally from the left via the standard GNS-action. The unbounded operator we will have need for is then given by This unbounded operator is selfadjoint and densely defined on H τ ⊕ H τ . Indeed (A s θ , H τ ⊕ H τ , D, Γ) defines a graded spectral triple for s ≥ 1, which we will use in Section 6. From now on, we write H for H τ ⊕ H τ . Note that since D is selfadjoint, the derivation δ := [D, −] : A s θ → B(H) satisfies δ(a * ) = −δ(a) * . We will denote by δ(a) also the bounded extension of [D, a] whenever this is defined.

Noncommutative Wiener Algebras.
Central to the study of time-frequency analysis are the twisted group algebras for lattices in the time-frequency plane. This is because they allow for faithful representations as "algebras of time-frequency shifts on a lattice", as we shall see below.
Let α, β ∈ R \ {0} and let Λ = αZ × βZ be a lattice in R × R ∼ = R 2 . Further, let c be a continuous 2-cocycle with values in T. We may then consider l 1 (Λ, c), which is just the Banach space l 1 (Λ) together with a (twisted) multiplication ♮ given by and involution a * = (a * (λ)) given by We will consider twisted involution algebras related to the canonical derivations on the noncommutative 2-torus. This is intimately related to polynomial decay of the l 1 (Λ)sequences. Hence we want to look at the twisted weighted group subalgebras l 1 We put on these algebras the norms The radial polynomial weight r s (x, ω) = (1 + |x| + |ω|) s is well-behaved enough for l 1 s (Λ, c) to be an involutive Banach algebra, see [Lue09].
and T x : Note that both modulation and translation are unitary operators on L 2 (R). Modulation and translation satisfy the commutation relation If we let c denote the continuous 2-cocycle on R 2 defined by c((x, ω), (y, η)) = e 2πiy·ω for (x, ω), (y, η) ∈ R 2 , we get the commutation relation for the time-frequency shifts π(x, ω) It follows that for the lattice Λ ⊂ R × R, the mapping λ → π(λ) defines a projective representation of Λ on L 2 (R). This in turn gives a faithful [Rie88] nondegenerate involutive representation of the involutive Banach algebra l 1 s (Λ, c) by definining We define the noncommutative Wiener algebra W 1 s (Λ, c) as the image of l 1 s (Λ, c) under the mapping π : l 1 s (Λ, c) → B(L 2 (R)). That is If we on W 1 s (Λ, c) consider the norm we get that l 1 s (Λ, c) and W 1 s (Λ, c) are isometrically isomorphic as involutive Banach algebras. From now on, we will identify these two algebras.
The universal enveloping C * -algebra of l 1 the supremum ranging over all faithful involutive Banach algebra representations of l 1 s (Λ, c). As l 1 s (Λ, c) is the twisted group algebra of an abelian group, namely the lattice Λ, l 1 s (Λ, c) is amenable. So we may identify for the above representation π. Since W 1 s (Λ, c) is generated by two (noncommuting) unitaries, the universal enveloping algebra C * (Λ, c) is the noncommutative 2-torus. The noncommutativity parameter θ described in Section 4.1 is determined by the lattice Λ = αZ × βZ. Indeed θ = αβ. It is clear that W 1 s (Λ, c) corresponds exactly to the algebra A s θ of Section 4.1 for all s ≥ 0.
The derivations δ 1 and δ 2 of Section 4.1 now become Indeed, this just follows by that (x, ω) = (αn 1 , βn 2 ) for some (n 1 , n 2 ) ∈ Z 2 . This is simply another normalization of the derivations. From this, D and δ = [D, −] of Section 4.1 can be written down explicitly. The following proposition is then just a reformulation of Proposition 4.1.

A Smooth
and so the second term is bounded by assumption. Then it suffices to prove that the first term extends to a bounded operator. To this end, let z ∈ Dom(D) and y ∈ H be arbitrary, and let {y n } n be a sequence in Dom(D) with lim n→∞ y n = y. Then From this expression we see that for ad(D) j (a)(D + i) −j+1 to be defined it suffices that the operator D i a is defined for 0 ≤ i ≤ j. For x ∈ Dom(D j ) we can use the Leibniz rule for δ 1 and δ 2 to write out the expression D i ax. Remembering that δ 1 and δ 2 commute and that D is given by we get that D i ax is a column matrix consisting of linear combination of terms δ t 1 1 δ t 2 2 (a) · δ d 1 1 δ d 2 2 (x) with max(t 1 , t 2 ), max(d 1 , d 2 ) ≤ i. In particular we need this to be defined for i = j. It then suffices that δ j 1 δ j 2 (a) is defined. But by definition of W 1 s (Λ, c) this is the case whenever s ≥ j.
It remains to show that ad(D) j (a)(D+i) −j+1 leaves Dom(D) invariant for 1 ≤ j ≤ m−1 and a ∈ W 1 s (Λ, c) for s ≥ m. Since (D + i) −1 : H → Dom(D) is onto, the question can be reduced to the question of whether or not Dad(D) j (a)(D + i) −j is defined everywhere. Once again using the rewriting above we write As above, it suffices that the operator D i a is defined for 0 ≤ i ≤ j + 1. For x ∈ Dom(D j ) we can, by the same arguments as before, write D i ax as a column vector consisting of linear combinations of terms of the form δ t 1 1 δ t 2 2 (a) · δ d 1 1 δ d 2 2 (x), for max(t 1 , t 2 ) ≤ j + 1 and max(d 1 , d 2 ) ≤ j. It is then sufficient that the terms δ t 1 1 δ t 2 2 (a) is defined for t 1 = t 2 = j + 1. But by definition of W 1 s (Λ, c) this is the case whenever s ≥ j + 1. It is clear that the same arguments apply to a * as a * ∈ W 1 s (Λ, c) whenever a ∈ W 1 s (Λ, c). In particular the argument is now applicable to k, and so for s ≥ k it follows that W 1 s (Λ, c) ⊂ A k . From now on, the above A k and the associated operator * -algebras from Section 3 is what we will refer to as the smooth structure on C * (Λ, c).
The above construction is known as the maximal operator * -algebra associated to a spectral triple. In Section 6 we shall have need for a related concept, the minimal operator * -algebra associated to a spectral triple. However, we postpone the treatment until then.

Basic Definitions and Facts.
To describe smooth structure in noncommutative geometry, we will need some appropriate modules. It turns out the well-studied function spaces known as modulation spaces [Fei83], in particular the polynomially weighted Feichtinger algebras [Fei81], are perfect for this problem. If g ∈ L 2 (R) is a window function, we define the short-time Fourier transform of a function or distribution f by Now fix φ(t) = e −πt 2 , the Gaussian. To introduce our modules of interest we then define the weighted Feichtinger algebra M 1 s (R) as follows These function spaces contain functions with desirable decay conditions for both time and frequency simultaneously. The choice of the Gaussian φ to describe these spaces may seem artificial, but it is a nontrivial fact that defining the spaces by choosing any nonzero function in Feichtinger's algebra M 1 (R) := M 1 0 (R) yield the same spaces, and will give equivalent norms, see [Grö01].
Sitting inside every one of the above modulation spaces is the Schwartz class. In fact Since the Schwartz class S(R) is dense in L 2 (R), it follows that M 1 s (R) is dense in L 2 (R) for all s ≥ 0.
Given a lattice Λ = αZ × βZ, its adjoint lattice is given by The notion of the adjoint lattice Λ o for a lattice Λ allows us to get a useful formula for the sum of the product of two STFTs. This is called the fundamental identity of Gabor analysis, or FIGA for short.
Proposition 5.1. Let Λ = αZ × βZ ⊂ R 2 be a lattice. For f 1 , f 2 , g 1 , g 2 ∈ M 1 s (R), s ≥ 0, the following identity holds The following proposition is the key ingredient in demonstrating that modulation spaces describe a smooth structure on the noncommutative torus. Remember the action π of Λ on L 2 (R), and the action of C * (Λ, c) on L 2 (R), also denoted π.

5)
For g ∈ S(R) and (a(λ)) λ∈Λ ∈ S(Λ) we have λ∈Λ a(λ)π(λ)g ∈ S(R), and we have the norm estimate λ∈Λ a(λ)π(λ)g M 1 Gabor Frames. The existence of smooth enough generating sequences in the sense of Section 3 will in the current setting turn out to be a result about existence of multi-window Gabor frames with windows in suitable modulation spaces. To this end, we include a brief introduction to Gabor frames.
Definition 5.3. A Gabor system G(g; Λ) is a collection of time-frequency shifts of a function g of the form {π(λ)g|λ ∈ Λ}. We call it a Gabor frame for L 2 (R) if it is a frame for the Hilbert space L 2 (R). That is, if the following inequalities are satisfied for all f ∈ L 2 (R) Extending to the case where we have functions g 1 , . . . , g n ∈ L 2 (R), we define a multiwindow Gabor system by G(g 1 , . . . , g n ; Λ) := G(g 1 ; Λ) ∪ · · · ∪ G(g n ; Λ). We call it a multiwindow Gabor frame for L 2 (R) if there exist constants 0 < A ≤ B < ∞ such that for all f ∈ L 2 (R). Again, if A = B = 1 we call G(g 1 , . . . , g n ; Λ) a tight normalized multi-window Gabor frame.
Intimately related to Gabor frames are the coefficient mapping and the synthesis mapping A straightforward calculation shows that D g,Λ = C * g,Λ . These allow us to define the Gabor frame operator.
Definition 5.4. For a Gabor frame G(g; Λ) we define the Gabor frame operator S g,Λ by Likewise, given a multi-window Gabor frame G(g 1 , . . . , g n ; Λ), we define the multi-window Gabor frame operator S g 1 ,...,gn,Λ by Note that boundedness of the (multi-window) Gabor frame operator is guaranteed by the upper norm bounds in Equation (40) and Equation (41). The corresponding lower bound guarantees that the (multi-window) Gabor frame operator is invertible. Also, since S g,Λ = C * g,Λ • C g,Λ , the Gabor frame operator is positive and thus the multi-window Gabor frame operator is positive, too. Hence for a Gabor frame G(g; Λ) (resp. a multi-window Gabor system G(g 1 , . . . g n ; Λ)) the corresponding Gabor frame operator S g,Λ (resp. multiwindow Gabor frame operator S g 1 ,...gn,Λ ) is a bounded, positive, and invertible operator.
In [Lue09] the following important result was shown for GRS-weights, see [Grö07].
In [Lue09] this was done by completing M 1 v (R) to a Hilbert C * -module over C * (Λ, c) (we do this in Section 5.2), and then interpreting Hilbert C * -module frames [FL02] as Hilbert space frames for L 2 (R).
Polynomial weights are GRS-weights, and so for any s ≥ 0 we can guarantee the existence of (normalized tight) multi-window Gabor frames for L 2 (R) with atoms g 1 , . . . , g n in M 1 s (R). This also includes the Schwartz class S(R), which one might view as M 1 ∞ (R) by the aforementioned characterization of the Schwartz class in terms of modulation spaces.
Moreover, Gabor frames can be used to characterize certain modulation spaces. This is the content of the following simplified proposition from [FG97]: Proposition 5.6. Let g ∈ M 1 (R) and f ∈ L 2 (R). Assume further that G(g; Λ) is a Gabor frame for L 2 (R). Then f ∈ M 1 s (R) if and only if (V g f (λ)) λ∈Λ ∈ l 1 s (Λ).
Consequently, we have a characterization of Schwartz functions in terms of Gabor frames: Proposition 5.7. Let g ∈ (R) and f ∈ L 2 (R). Assume further that G(g; Λ) is a Gabor frame for L 2 (R). Then f ∈ S(R) if and only if (V g f (λ)) λ∈Λ ∈ S(Λ).

Modulation
Spaces as Modules over the Noncommutative 2-Torus. For the time being we will consider the modulation spaces as left modules over the noncommutative Wiener algebras. For this purpose the following definitions are natural. Let f ∈ M 1 s (R) and a = λ∈Λ a(λ)π(λ) ∈ W 1 s (Λ, c). Then we define The action is well-defined and bounded by Proposition 5.2. We obtain a natural left W 1 s (Λ, c)-valued inner product on M 1 s (R) in the following way: Let f, g ∈ M 1 s (R). Then define The fact that this defines an element of W 1 s (Λ, c), is guaranteed by Proposition 5.2. For notational ease we will denote the inner product by W 1 s ., . from now on. The left action and the inner product are compatible in the following sense. The proof is essentially identical to the one in [Lue09], but we reiterate it here.
Proposition 5.8. M 1 s (R) is a left inner product W 1 s (Λ, c)-module for the left action of W 1 s (Λ, c) given by (48) a · g = λ∈Λ a(λ)π(λ)g for a = (a(λ)) λ∈Λ ∈ W 1 s (Λ, c) and g ∈ M 1 s (R), and the W 1 s (Λ, c)-valued inner product Proof. We already know the action is bounded, and C-linearity is obvious. Now note that c(−λ, µ)c(λ, −λ + µ) = c(−λ, λ) = c(λ, λ). Then we have Antisymmetry of the inner product, that is, W 1 s f, g = W 1 s g, f * follows from the computation: Lastly, we need to verify positive definiteness. This will follow by the fundamental identity of Gabor analysis, see Proposition 5.1. Note that since the representation of W 1 s (Λ, c) is faithful on L 2 (R) and * -homomorphisms preserve positivity, it suffices to check that check positivity for f in this dense subspace. To this end, let f, g ∈ M 1 s (R). Then All in all this makes M 1 s (R) into an inner product module over W 1 s (Λ, c). We may complete both the pre-C * -algebra and the inner product module to obtain a left Hilbert C * -module over the noncommutative torus C * (Λ, c). Denote the module completion by Λ V , and denote the extension of the inner product by Λ ., . . We will also denote W 1 s ., . by Λ ., . from now on. By arguments in [Lue09] Λ V actually becomes a full Hilbert C * -module over C * (Λ, c).
Since the canonical trace tr : C * (Λ, c) → C from Section 4.1 is continuous, and for all g ∈ M 1 s (R), we obtain the following.
In particular, elements of the modules can still be regarded as L 2 -functions, which we will have need for in Section 5.3.

Modulation Spaces as Smooth
Modules. At last we are in a position to demonstrate that modulation spaces are examples of smooth modules over C * (Λ, c). This is the content of the following theorem, which is our first main result. Recall that M 1 ∞ (R) denotes the Schwartz space S(R).
Theorem 5.10. Let C * (Λ, c) be given the smooth structure {A k } as in Section 4.3 and let s ∈ [1, ∞]. A tight normalized multi-window Gabor frame G(g 1 , . . . , g n ; Λ) for L 2 (R), with g 1 , . . . g n ∈ M 1 s (R), gives Λ V the structure of a C k -module over C * (Λ, c), where k = ⌊s⌋. We have adopted the convention ⌊∞⌋ = ∞. In particular S(R) gives Λ V the structure of a smooth module over C * (Λ, c).
Proof. Given s ∈ [1, ∞], consider a tight normalized multi-window Gabor frame G(g 1 , . . . , g n ; Λ) for L 2 (R) with g i ∈ M 1 s (R) for i = 1, . . . , n. The existence of such a multi-window Gabor frame is guaranteed by Proposition 5.5. Then for any f ∈ L 2 (R) we have We may then reformulate this in terms of the multi-window Gabor frame operator in the following way = (S g 1 ,...,gn,Λ f |f ).
By the polarization identity this implies that S g 1 ,...,gn,Λ is the identity on L 2 (R). Since by Proposition 5.9 Λ V ⊂ L 2 (R), the multi-window Gabor frame operator S g 1 ,...,gn,Λ is also the identity on Λ V . We may now reformulate this in terms of module operators. For f ∈ M 1 s (R) we have where the operator g i ⊗ g i : f → Λ f, g i g i is a "rank one operator" on the Hilbert C *module Λ V . We may extend this by continuity to the whole module Λ V . Since this now holds for all f ∈ Λ V and the action of C * (Λ, c) on Λ V is faithful, we get In particular, Λ V is a finitely generated module, generated by g 1 , . . . , g n . So when considering the smooth submodules from Equation (7) the norm condition in H A k is irrelevant. The same is true for the norms on the matrices in Definition 3.3. Since by Proposition 5.2 and Proposition 4.3 Λ g m , π(λ)g l ∈ W 1 s (Λ, c) ⊂ A k for k = ⌊s⌋, and all m, l ∈ {1, . . . , n}, this completes the proof.
Since multiwindow Gabor frames can be viewed as generators of the Hilbert C * -module Λ V , the preceding statement shows that Gabor frames in M 1 s (R) have more regularity for increasing s. This turns the naive point of view that Gabor atoms in M 1 s (R) are more regular the larger s is into a rigorous statement.

Modulation Spaces as Operator * -Correspondences
Already having demonstrated that a modulation space M 1 s (R), s ∈ [0, ∞], is an inner product module over the noncommutative Wiener algebra W 1 s (Λ, c), we can prove that M 1 s (R) is an operator * -correspondence in the language of [BKM18] and [Kaa16]. More precisely, a suitable completion of M 1 s (R) becomes an operator * -correspondence, in a sense that will be described in Section 6.1 and Section 6.2. This will further allow us to represent the completion of M 1 s (R) as a corner in a linking operator * -algebra. However, the completion of M 1 s (R) we will need is not the same we have used so far. Specifically, we have up to this point used the maximal operator * -algebra related to a spectral triple. As alluded to earlier, we will have need for the minimal operator * -algebra related to a spectral triple, and so we will discuss this in Section 6.2.
Before going further, we are however going change the framework to consider right modules rather than left modules. We do this because operator * -correspondences and C * -correspondences are typically formulated for right modules. To see how to do this, we include the following lemma from [Lue09].
Lemma 6.1. Let A be a C * -algebra and ( A V, A ., . ) be a left Hilbert C * -module. Then the opposite module V op is a right Hilbert C * -module for the opposite algebra A op with A op -valued inner product ., .
It is further known that the opposite algebra of C * (Λ o , c) is C * (Λ o ,c) [Rie88]. In fact, Λ V as above becomes a C * (Λ, c)-C * (Λ o ,c)-imprimitivity bimodule if the right action is normalized correctly. We shall however not have need for this structure. To make precise the module structure we cite the following (simplified) theorem from [Lue09], where we have skipped the normalizations that would produce an imprimitivity bimodule. Note that we still keep the C-valued inner product linear in the first argument.
Theorem 6.2. Let Λ = αZ × βZ be a lattice in R 2 . Then M 1 s (R) completes to a full right when completed with respect to the norm ||f || From now on, this is the module structure we will use. Note that the derivations and the spectral triple from Section 4.1 carry over in a natural way.
6.1. Operator *-Correspondences. The notion of an operator * -correspondence will allow us to represent modulation spaces M 1 s (R), or rather, the completion M s,∇ introduced in Section 6.2 below, as a corner in an operator linking algebra. Let us first define the relevant terms. Definition 6.3. Let B and A be operator * -algebras and let M be a right operator *module over A. We call M an operator * -correspondence from B to A when there is a left operator B-module structure on M such that Furthermore, if M and N are operator * -correspondences from B to A and from D to C, respectively, we say M and N are cb-isomorphic if there exists a completely bounded operator space isomorphism φ : M → N together with completely bounded isomorphisms of operator * -algebras β : B → D and α : A → C such that iii) φ(b · x) = β(b) · φ(x) and φ(x · a) = φ(x) · α(a) for all b ∈ B, x ∈ M, and a ∈ A. iv) φ(x), φ(y) N = α( x, y M ) for all x, y ∈ M.
This generalizes C * -correspondences. For C * -correspondences it is always true that given a right Hilbert C * -module X over a C * -algebra A, X becomes a K(X)-A-correspondence, where K(X) are the usual "compact operators" on X. Indeed the analogous statement is true for operator * -correspondences. Given a right operator * -module M over an operator * -algebra A, M becomes an operator * -correspondence between K(M) and A, where K(M) is the operator * -algebra of "compact operators" on M, see [BLM04], [BKM18]. We shall however not have need for K(M).
It is known that C * -correspondences and the respective C * -algebras can be embedded in a C * -algebra respecting the bimodule structure and the hermitian structure. The analogous statement is in fact true for operator * -correspondences. Operator * -correspondences and the respective operator * -algebras can be embedded in an operator * -algebra up to completely bounded isomorphism respecting the module structure, as well as the involutive structure and the inner product structure up to conjugation by a selfadjoint unitary. The following theorem from [BKM18] contains all relevant results we will need.
for all b ∈ B, a ∈ A, x ∈ M. iii) U implements the involutive structure and A-valued inner product in the following sense: for all b ∈ B, x, y ∈ M, a ∈ A. Here * refers both to the involutions in B and A, as well as the Hilbert space adjoint operation in B(H φ ).
We remark that in [BKM18] it is proved that even more can be said. In particular, operator * -correspondences can be represented as corners in operator linking algebras. This gives a particularly simple representation to work with. 6.2. An Operator *-Correspondence Structure on Modulation Spaces. At last we have everything needed to represent modulation spaces as subspaces of an operator * -algebra. In this section we fix s ∈ [1, ∞].
We consider the spectral triple given by (W 1 s (Λ o ,c), H, D), with H and D constructed as in Section 4.1 and Section 4.2, but for C * (Λ o ,c). The derivation δ = [D, −] : W 1 s (Λ o ,c) → B(H) is closable, which can be seen by the following argument. Let {a n } n be a sequence in W 1 s (Λ o ,c) such that lim n→∞ a n 0 0 a n = 0 in B(H), while lim n→∞ [D, a n ] = T ∈ B(H). It suffices to prove that T = 0. For this purpose, let g, h ∈ Dom(D). Then, since D is adjointable (even selfadjoint) with respect to the inner product on H, we have T g, h = lim n→∞ [D, a n ]g, h = lim n→∞ (Da n − a n D)g, h = lim n→∞ ( Da n g, h − a n Dg, h ) = lim n→∞ ( a n g, Dh − a n Dg, h ) = 0g, Dh − 0Dg, h = 0.
Since g and h were arbitrary in Dom(D) and Dom(D) is dense in H, this proves the assertion. Hence we may take the closure and we denote by A s,δ the closure of W 1 s (Λ o ,c) in the graph norm. We obtain an operator * -algebra structure on A s,δ by putting on it the norms induced by the injective * -homomorphism where we have suppressed that a acts diagonally on H.
Proof. This is just the calculation Hence no further completion is required on A s,δ . The structure induced by ρ above is the minimal operator * -algebra associated to the spectral triple (W 1 s (Λ o ,c), H, D). Denote by A the C * -algebra obtained as the norm closure of A s,δ in B(H). Since Λ o is an abelian group, hence amenable, it follows that A = C * (Λ o ,c). Further, denote by Ω(A s,δ ) the smallest C * -subalgebra of B(H) such that (56) 1 ∈ Ω(A s,δ ), A ⊂ Ω(A s,δ ), and δ(a) ∈ Ω(A s,δ ) for all a ∈ A s,δ .
Then, since δ is the extension of the operator which sends a ∈ A s,δ to the operator in B(H) given by left multiplication by the matrix (57) 0 δ 1 (a) + iδ 2 (a) −δ 1 (a) −īδ 2 (a) 0 we see that Ω(A s,δ ) ∼ = C * (Λ o ,c)⊕C * (Λ o ,c). In the sequel these will be used interchangeably to sometimes shorten notation. In Section 5.2 we saw that M 1 s (R) becomes a left inner product module over W 1 s (Λ o ,c). Switching to right modules as described above, completing M 1 s (R) with respect to the induced norm from A s,δ and extending the action in the usual way, we obtain a right Banach * -module M s,δ over A s,δ . Then M s,δ is dense in V Λ o and f, g Λ o ∈ A s,δ for all f, g ∈ M s,δ . Note that this is not yet an operator * -module over A s,δ with respect to the operator * -algebra structure described above, but we will fix this below.