Gabor duality theory for Morita equivalent C∗-algebras

The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalence bimodules with some extra properties. For certain twisted group [Formula: see text]-algebras, the reformulation of the duality principle to the setting of Morita equivalence bimodules reduces to the well-known Gabor duality principle by localizing with respect to a trace. We may lift all results at the module level to matrix algebras and matrix modules, and in doing so, it is natural to introduce [Formula: see text]-matrix Gabor frames, which generalize multi-window super Gabor frames. We are also able to establish density theorems for module frames on equivalence bimodules, and these localize to density theorems for [Formula: see text]-matrix Gabor frames.

Invariant States on Noncommutative Tori

For any number $h$ such that $\hbar :=h/2\pi $ is irrational and any skew-symmetric, non-degenerate bilinear form $\sigma :{{\mathbb{Z}}}^{2g}\times{{\mathbb{Z}}}^{2g} \to{{\mathbb{Z}}}$, let be ${{\mathcal{A}}}^h_{g,\sigma }$ be the twisted group *-algebra ${{\mathbb{C}}}[{{\mathbb{Z}}}^{2g}]$ and consider the ergodic group of *-automorphisms of ${{\mathcal{A}}}^h_{g,\sigma }$ induced by the action of the symplectic group $\textrm{Sp} \,({{\mathbb{Z}}}^{2g},\sigma )$. We show that the only $\textrm{Sp} \,({{\mathbb{Z}}}^{2g},\sigma )$-invariant state on ${{\mathcal{A}}}^h_{g,\sigma }$ is the trace state $\tau $.