We investigate and review how Fourier transform is involved in the analysis of a twisted group algebra $L^1(G, \sigma)$ for $G=\widehat{ \Gamma}\times \Gamma$ and $\sigma:G\times G \to \mathbb{ T}$ 2- cocycle where $\Gamma$ is a locally compact abelian group and $\widehat{ \Gamma}$ its Pontryagin dual. By weaving the Schr\"{ o}dinger representation and Fourier transform, we construct the dual equivalence bimodule of the Heisenberg bimodule generated by the dual Schr\"{ o}dinger representation and observe several relations between them including the application of noncommutative solitons.
Categories: math.OA math-ph math.MP