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

Abstract

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 ,

Spectral invariance of $*$-representations of twisted convolution algebras with applications in Gabor analysis

“…, g k ; ∆) as in Remark 4.4 below [26, Lemma 4.9]. The same is true if we consider matrix frames introduced in[3], see[3, Proposition 4.34].…”

“…, S −1/2 g k ∈ S 0 (G). Indeed one can go even further and do this for the matrix Gabor frames introduced in [3], which generalize multi-window super Gabor frames, using the setup from the same article. The key observation for doing this is that since ℓ 1 (∆ • , c) is spectrally invariant in B(L 2 (G)) we also have that M n (ℓ 1 (∆ • , c)) is spectrally invariant in M n (B(L 2 (G))) for any n ∈ N [42].…”

“…We will do this by restating the problem in operator algebraic terms and then use Theorem 3.1. Exploring the interplay between Gabor analysis and operator algebras has gained much popularity in recent years [2,3,11,26,28,34,35]. The field of Gabor analysis has its origins in the seminal paper of Gabor [14], where he claimed that it is possible to obtain basis-like representations of functions in L 2 (R) in terms of the set {e 2πilx φ(x − k) : k, l ∈ Z}, where φ denotes a Gaussian.…”

Modular Riesz bases versus Riesz bases in Hilbert C∗-modules

“…The theory of frames in Hilbert C * -modules was introduced by Frank and Larson [6]; more recent works include [3,5,7,8,11,14]. Hilbert C * -modules are generalizations of Hilbert spaces in which the inner product takes values in a C * -algebra.…”