Weak frames in Hilbert C∗-modules with application in Gabor analysis

Abstract

In the first part, we describe the dual ${\ell }^2(\mathsf{A}{)}^{\prime }$ of the standard Hilbert $C^{\ast }$-module ${\ell }^2(\mathsf{A})$ over an arbitrary (not necessarily unital) $C^{\ast }$-algebra $\mathsf{A}$. When $\mathsf{A}$ is a von Neumann algebra, this enables us to construct explicitly a self-dual Hilbert $\mathsf{A}$-module ${\ell }_{\mathrm{strong}}^2(\mathsf{A})$ that is isometrically isomorphic to ${\ell }^2(\mathsf{A}{)}^{\prime }$, which contains ${\ell }^2(\mathsf{A})$, and whose $\mathsf{A}$-valued inner product extends the original inner product on ${\ell }^2(\mathsf{A})$. This serves as a concrete realization of a general construction for Hilbert $C^{\ast }$-modules over von Neumann algebras introduced by Paschke. Then we introduce a concept of a weak Bessel sequence and a weak frame in Hilbert $C^{\ast }$-modules over von Neumann algebras. The dual ${\ell }^2(\mathsf{A}{)}^{\prime }$ is recognized as a suitable target space for the analysis operator. We describe fundamental properties of weak frames such as the correspondence with surjective adjointable operators, the canonical dual, the reconstruction formula, and so on, first for self-dual modules and then, working in the dual, for general modules. The last part describes a class of Hilbert $C^{\ast }$-modules over $L^{\infty }(I)$, where $I$ is a bounded interval on the real line, that appears naturally in connection with Gabor (i.e., Weyl–Heisenberg) systems. We then demonstrate that Gabor Bessel systems and Gabor frames in $L^2(\mathbb{R})$ are in a bijective correspondence with weak Bessel systems and weak frames of translates by $a$ in these modules over $L^{\infty }[0,\frac{1}{b}]$, where $a,b>0$ are the lattice parameters. In this setting new proofs of several classical results on Gabor frames are demonstrated and some new ones are obtained.