The duality principle for Gabor frames is one of the pillars of Gabor
analysis. We establish a far-reaching generalization to Morita equivalent
Cβ-algebras where the equivalence bimodule is a finitely generated
projective Hilbert Cβ-module. These Hilbert Cβ-modules are equipped with
some extra structure and are called Gabor bimodules. We formulate a duality
principle for standard module frames for Gabor bimodules which reduces to the
well-known Gabor duality principle for twisted group Cβ-algebras of a
lattice in phase space. We lift all these results to the matrix algebra level
and in the description of the module frames associated to a matrix Gabor
bimodule we introduce (n,d)-matrix frames, which generalize superframes and
multi-window frames. Density theorems for (n,d)-matrix frames are
established, which extend the ones for multi-window and super Gabor frames. Our
approach is based on the localization of a Hilbert Cβ-module with respect to
a trace.Comment: 36 page