Gabor duality theory for Morita equivalent C∗-algebras

Are Austad,Franz Luef,Mads S Jakobsen

doi:10.1142/s0129167x20500731

Are Austad, Franz Luef + Show 1 more

Open Access

https://doi.org/10.1142/s0129167x20500731

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Abstract

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.

Highlights

Hilbert C∗-modules are well-studied objects in the theory of operator algebras and Rieffel made the crucial observation that they provide the correct framework for the extension of Morita equivalence of rings to C∗-algebras
For certain twisted group C∗-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 are able to establish density theorems for module frames on equivalence bimodules, and these localize to density theorems for (n, d)-matrix Gabor frames

Summary

Introduction

Hilbert C∗-modules are well-studied objects in the theory of operator algebras and Rieffel made the crucial observation that they provide the correct framework for the extension of Morita equivalence of rings to C∗-algebras. Luef observed that finitely generated equivalence bimodules could be described in terms of finite standard module frames He used this in his study of Heisenberg modules — a class of projective Hilbert C∗-modules over twisted group C∗-algebras. Since we aim to mimic the situation of Gabor analysis, the positive linear functional that we localize our Morita equivalence bimodule with respect to will have. The following was proved in [3, Proposition 3.1] and ensures that this procedure works Note that if both C∗-algebras are unital the induced trace is a finite trace as in [26], the result can be deduced from [26, Proposition 2.2].

The equivalence bimodule picture

Passing to the localization

The Link to Gabor Analysis