Frames and outer frames for Hilbert C^*-modules

TL;DR

This paper extends the theory of frames to Hilbert C*-modules over arbitrary C*-algebras, introducing outer frames and establishing a comprehensive framework for duals, perturbations, and extensions.

Contribution

It introduces the concept of outer frames for non-unital C*-algebra modules and establishes a bijective correspondence between frames and certain surjections, unifying the theory.

Findings

Outer frames generalize frames in non-unital modules.

A bijective correspondence exists between frames and adjointable surjections.

New results on dual frames, perturbations, and finite extensions.

Abstract

The goal of the present paper is to extend the theory of frames for countably generated Hilbert $C^*$-modules over arbitrary $C^*$-algebras. In investigating the non-unital case we introduce the concept of outer frame as a sequence in the multiplier module $M(X)$ that has the standard frame property when applied to elements of the ambient module $X$. Given a Hilbert $\A$-module $X$, we prove that there is a bijective correspondence of the set of all adjointable surjections from the generalized Hilbert space $\ell^2(\A)$ to $X$ and the set consisting of all both frames and outer frames for $X$. Building on a unified approach to frames and outer frames we then obtain new results on dual frames, frame perturbations, tight approximations of frames and finite extensions of Bessel sequences.