Weak frames in Hilbert \(C^*\)-modules with application in Gabor analysis (Q2329258)

Given a \(C^\ast\) algebra \(A\) the author considers the (left) Hilbert \(C^\ast\)-module \(\ell^2(A)\) defined as \[ \ell^2(A) = \{(a_n)_n:\ a_n\in A,\ \sum_{n=1}^\infty a_n a_n^\ast \ \mbox{converges in norm in}\ A\}. \] The goal of section 2 is to describe the adjointable dual \(\ell^2(A)^\ast\) and the dual \(\ell^2(A)^\prime,\) consisting on those bounded module maps from \(\ell^2(A)\) into \(A.\) In Theorem 2.3, \(\ell^2(A)^\prime\) is identified with \[ \ell^2_{\mathrm{strong}}(LM(A)) = \{(c_n)n:\ c_n\in LM(A),\ \sup_N\|\sum_{n=1}^N c_n c_n^\ast\| < \infty\}, \] where \(LM(A)\) stands for the set of left multipliers of \(A.\) The adjointable dual \(\ell^2(A)^\ast\) coincides with the set \(M\left(\ell^2(A)\right)\) of multipliers of \(\ell^2(A).\) If \(A\) is a von Newmann algebra then \(\ell^2(A)^\prime\) is antilinearly isometrically isomorphic to the Hilbert \(A\)-module \(\ell^2_{\mathrm{strong}}(A),\) and it is self-dual. In section 3, the author introduces the concept of a weak Bessel sequence and a weak frame in a self-dual Hilbert \(C^\ast\)-module \(X\) over a von Newmann algebra \(A.\) The corresponding analysis operator \(U:X\to \ell^2_{\mathrm{strong}}(A)\) and its adjoint, the synthesis operator, are discussed in Theorem 3.4. The invertibility of \(U^\ast U\) is the content of Theorem 3.15. Section 4 discusses a sufficient condition on a sequence \(\left(x_n\right)_n,\) described in terms of Gram matrices, to ensure the weak Bessel property under the additional condition that the von Newmann algebra \(A\) is commutative. Section 5 describes a class of Hilbert \(C^\ast\)-modules that are naturally connected to Gabor analysis. Let \(L^\infty_a\left(\ell^2\right)\) denote the set of those measurable functions \(f:{\mathbb R}\to {\mathbb C}\) such that \(\mbox{ess}\sup_{x\in [0,a]}\sum_{n\in {\mathbb Z}}|f(x-na)|^2 < \infty.\) \(L^\infty_a\left(\ell^2\right)\) is a self-dual Hilbert \(C^\ast\)-module over the von Newmann algebra \(L^\infty[0,a]\) when it is endowed with the vector valued inner product \[ L^\infty_a\left(\ell^2\right)\times L^\infty_a\left(\ell^2\right)\to L^\infty[0,a],\ \ (f,g)\mapsto \sum_{n\in {\mathbb Z}}f(x-na)\overline{g(x-na)} \] (Theorem 5.6). A correspondence between Gabor Bessel sequences (Gabor frames) in \(L^2({\mathbb R})\) and weak Bessel sequences (weak frames) of translates in the Hilbert \(C^\ast\)-modules \(L^\infty_a\left(\ell^2\right)\) is established in Theorem 5.9. This correspondence permits to reinterpret in section 6 some of the classical results from Gabor analysis. In particular, a Walnut representation of the frame operator of Gabor Bessel sequences in \(L^2({\mathbb R})\) is obtained without the assumption that the windows belong to the Wiener space or satisfy the CC condition.