Frames in Hilbert C*-modules and C*-algebras

TL;DR

This paper develops a comprehensive modular frame theory within C*-algebras and Hilbert C*-modules, utilizing geometric dilation and projections to establish frame representations, with applications to various algebraic and analytical structures.

Contribution

It introduces a unified approach to frames in Hilbert C*-modules, extending classical frame theory and connecting it with operator algebra structures and bundle theory.

Findings

Established frame representations and decomposition theorems.

Derived similarity and equivalence results for frames.

Applied results to Cuntz-Krieger-Pimsner algebras and wavelet frames.

Abstract

We present a general approach to a modular frame theory in C*-algebras and Hilbert C*-modules. The investigations rely on the idea of geometric dilation to standard Hilbert C*-modules over unital C*-algebras that possess orthonormal Hilbert bases, and of reconstruction of the frames by projections and by other bounded modular operators with suitable ranges. We obtain frame representations and decomposition theorems, as well as similarity and equivalence results for frames. Hilbert space frames and quasi-bases for conditional expectations of finite index on C*-algebras appear as special cases. Using a canonical categorical equivalence of Hilbert C*-modules over commutative C*-algebras and (F)Hilbert bundles the results find a reintepretation for frames in vector and (F)Hilbert bundles. Fields of applications are investigations on Cuntz-Krieger-Pimsner algebras, on conditional expectations of finite index, on various ranks of C*-algebras, on classical frame theory of Hilbert spaces (wavelet and Gabor frames), and others. 2001: In the introduction we refer to related publications in detail.