  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.
