Modular frames for Hilbert C*-modules and symmetric approximation of frames
Michael Frank, David R. Larson
TL;DR
The paper extends classical frame theory to Hilbert C*-modules by introducing modular frames over unital C*-algebras, establishing a frame transform $\theta$ and frame operator $S=(\theta^*\theta)^{-1}$ that enable a reconstruction formula $x=\sum_j \langle x,x_j\rangle S(x_j)$. It proves the existence of standard normalized tight modular frames for finitely or countably generated modules via embeddings into $A^n$ or $l_2(A)$, and develops invariants for finitely generated projective modules using Gram data $a_{ij}=\langle x_i,x_j\rangle$, showing these determine the module up to isomorphism. The authors also resolve an operator-theoretic problem of representing sequences with $\mathrm{id} = \sum b_i^* b_i$ in $B(l_2)$ through modular-frame machinery, providing a detailed decomposition involving projections and partial isometries. Finally, the work analyzes symmetric (normalized) tight-frame approximation, showing that the closest tight frame to a given frame in the operator-norm sense is connected to $S^{1/2}$ via Löwdin-type orthogonalization in commutative settings, while noncommutative generalizations remain open. The results bridge modular frames with operator-algebraic structures and extend key aspects of Hilbert-space frame theory to the modular setting, with implications for wavelet/frame theory and related applications.
Abstract
We give a comprehensive introduction to a general modular frame construction in Hilbert C*-modules and to related modular operators on them. The Hilbert space situation appears as a special case. The reported investigations rely on the idea of geometric dilation to standard Hilbert C*-modulesover unital C*-algebras that admit an orthonormal Riesz basis. Interrelations and applications to classical linear frame theory are indicated. As an application we describe the nature of families of operators {S_i} such that SUM_i S*_iS_i=id_H, where H is a Hilbert space. Resorting to frames in Hilbert spaces we discuss some measures for pairs of frames to be close to one another. Most of the measures are expressed in terms of norm-distances of different kinds of frame operators. In particular, the existence and uniqueness of the closest (normalized) tight frame to a given frame is investigated. For Riesz bases with certain restrictions the set of closetst tight frames often contains a multiple of its symmetric orthogonalization (i.e. Löwdin orthogonalization).