A module frame concept for Hilbert C*-modules

TL;DR

This paper introduces a general theory of module frames in Hilbert C*-modules, extending classical Hilbert space frame concepts to the setting of C*-algebras with new representation and decomposition results.

Contribution

It develops a comprehensive module frame theory in Hilbert C*-modules, including representation, decomposition, and equivalence theorems, generalizing known Hilbert space results.

Findings

Frame representation and decomposition theorems established

Similarity and equivalence of frames analyzed

Generalization of Casazza's results to C*-module setting

Abstract

The goal of the present paper is a short introduction to a general module frame theory in C*-algebras and Hilbert C*-modules. The reported investigations rely on the idea of geometric dilation to standard Hilbert C*-modules over unital C*-algebras that possess orthonormal bases, and of reconstruction of the frames by projections and other bounded module operators with suitable ranges. We obtain frame representation and decomposition theorems, as well as similarity and equivalence results. The relative position of two and more frames in terms of being complementary or disjoint is investigated in detail. In the last section some recent results by P. G. Casazza are generalized to our setting. The Hilbert space situation appears as a special case. For detailled proofs we refer to another paper also contained in the ArXiv.