The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalence bimodules with some extra properties. For certain twisted group [Formula: see text]-algebras, the reformulation of the duality principle to the setting of Morita equivalence bimodules reduces to the well-known Gabor duality principle by localizing with respect to a trace. We may lift all results at the module level to matrix algebras and matrix modules, and in doing so, it is natural to introduce [Formula: see text]-matrix Gabor frames, which generalize multi-window super Gabor frames. We are also able to establish density theorems for module frames on equivalence bimodules, and these localize to density theorems for [Formula: see text]-matrix Gabor frames.

Let be a closed, cocompact subgroup of G × G, where G is a second countable, locally compact abelian group. Using localization of Hilbert C *-modules, we show that the Heisenberg module E (G) over the twisted group C *-algebra C * (, c) due to Rieffel can be continuously and densely embedded into the Hilbert space L 2 (G). This allows us to characterize a finite set of generators for E (G) as exactly the generators of multi-window (continuous) Gabor frames over , a result which was previously known only for a dense subspace of E (G). We show that E (G) as a function space satisfies two properties that make it eligible for time-frequency analysis: Its elements satisfy the fundamental identity of Gabor analysis if is a lattice, and their associated frame operators corresponding to are bounded.

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For a 2nd-countable locally compact Hausdorff étale groupoid ${\mathcal{G}}$ with a continuous $2$-cocycle $\sigma $ we find conditions that guarantee that $\ell ^1 ({\mathcal{G}},\sigma )$ has a unique $C^*$-norm.

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.Proposition 2.1 ([24], Theorem 2.1.11). Let A be a unital C * -algebra and let B be aDefinition 2.2. Let A be a unital C * -algebra and let B ⊂ A be a Banach * -subalgebra of A with the same unit. We say thatNow recall that a selfadjoint element a in a C * -algebra with σ A (a) ⊂ [0, ∞) is called positive. We state a useful characterization of positivity. Proposition 2.3. Let A be a C * -algebra. For a = a * ∈ A we have σ A (a) ⊂ [0, ∞) if and only if a = b * b for some b ∈ A.Denote by A + the set of positive elements in the C * -algebra A. The positive elements form a cone. In particular, if a ∈ A + then ka ∈ A + for all k ∈ [0, ∞), and if a 1 , a 2 ∈ A + then a 1 + a 2 ∈ A + . We also obtain a partial order on A + by a ≤ b if and only if b − a ∈ A + . Note that not all elements of A + are comparable, but all elements are comparable to 1 A in the case A is unital.Central to our results in Section 3 will be the localization of a Hilbert C * -module. For this we need positive linear functionals.Definition 2.4. A positive linear functional on a C * -algebra A is a linear functional φ such that φ(A + ) ⊂ [0, ∞). If φ = 1 we say φ is a state. Remark 2.5. If φ : A → C is a positive linear functional and A is unital, it is known that φ is a state if and only if φ(1 A ) = 1.We will denote the set of adjointable operators on the Hilbert A-module E by End A (E), and the set of compact module operators by K(E). The following two results will be of great importance in our approach to duality theorems.Proposition 2.6 ([20], Proposition 1.1). Let A be a C * -algebra. If E is an inner product A-module and f, g ∈ E, then A g, f A f, g ≤ A f, f A g, g ,