| Volume 14, Number 5, 2024, Pages - DOI：10.11948/JAAC-2023-0242 |
| Universal approach to the Takesaki-Takai $\gamma $-duality for crossed products |
| Mykola Ivanovich Yaremenko |
| Keywords:Takai Duality, $\gamma $-duality, Wigner function, \textbf{\textit{$C^{*} $}}-algebra, Pontryagin duality, induced representation, cross product |
| Abstract: |
| In this article, we generalize and simplify the proof of the Takesaki-Takai $\gamma $-duality theorem.
Assume a morphism \textbf{\textit{$\omega \; :\; G\to Aut\left({\rm A}\right)$}} is a projective representation of the locally compact Abel group \textbf{\textit{$G$}} in \textbf{\textit{$Aut\left({\rm A}\right)$}}, mapping $\gamma \; :\; G\to G$ is continuous, and $\left({\rm A},\; G,\; \omega \right)$ is a dynamic system then there exists isomorphism \[\Upsilon \; :\; Env_{\hat{\omega }} {}^{\gamma } \left(L^{1} \left(\hat{G},\; Env_{\omega } {}^{\gamma } \left(L^{1} \left(G,\; {\rm A}\right)\right)\right)\right)\to {\rm A}\otimes LK\left(L^{2} \left(G\right)\right) \] which is the equivariant for the double dual action \[\hat{\hat{\omega }}\; :\; G\to Aut\left(Env_{\hat{\omega }} {}^{\gamma } \left(L^{1} \left(\hat{G},\; Env_{\omega } {}^{\gamma } \left(L^{1} \left(G,\; {\rm A}\right)\right)\right)\right)\right).\]
These results deepen our understanding of the representation theory and are especially interesting given their possible applications to problems of the quantum theory. |
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