Frames and outer frames for Hilbert C*-modules

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 l(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.