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

Abstract: In the first part of the paper we describe the dual \ell^2(A)^{\prime} of the standard Hilbert C*-module \ell^2(A) over an arbitrary (not necessarily unital) C*-algebra A. When A is a von Neumann algebra, this enables us to construct explicitly a self-dual Hilbert A-module \ell^2_{\text{strong}}(A) that is isometrically isomorphic to \ell^2(A)^{\prime}, which contains \ell^2(A), and whose A-valued inner product extends the original inner product on \ell^2(A). This serves as a concrete realization of a general construction for Hilbert C*-modules over von Neumann algebras introduced by W. Paschke. Then we introduce a concept of a weak Bessel sequence and a weak frame in Hilbert C*-modules over von Neumann algebras. The dual \ell^2(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, etc; first for self-dual modules and then, working in the dual, for general modules. In the last part of the paper we describe a class of Hilbert C*-modules over L^{\infty}(I), where I is a bounded interval on the real line, that appear naturally in connection with Gabor (i.e. Weyl-Heisenberg) systems. We then demonstrate that Gabor Bessel systems and Gabor frames in L^2(\Bbb R) are in a bijective correspondence with weak Bessel systems and weak frames of translates by a in these modules over L^{\infty}[0,1/b], where a,b>0 are the lattice parameters. In this setting some well known results on Gabor systems are discussed and some new are obtained.