A module frame concept for Hilbert C*-modules

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 Hilbert 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 some detail. In the last section some recent results of P. G. Casazza are generalized to our setting. The Hilbert space situation appears as a special case. For the details of most of the proofs we refer to our basic publication [8]. Frames serve as a replacement for bases in Hilbert spaces that guarantee canonical reconstruction of every element of the Hilbert space by the reconstruction formula, however, giving up linear independence of the elements of the generating frame sequence. They appear naturally as wavelet generated and Weyl-Heisenberg / Gabor frames since often sequences of this type do not become orthonormal or Riesz bases, [11, 2, 12]. Similarly, the concept of module frames has become a 1991 Mathematics Subject Classification. Primary 46L99; Secondary 42C15, 46H25, 47A05.