UNIVERSAL APPROACH TO THE TAKESAKI-TAKAI $\gamma $-DUALITY

In this article, we generalize and simplify the proof of the Takesaki-Takai $ \gamma $-duality theorem. Assume a morphism $ \omega \; :\; G\to Aut\left({\rm A}\right) $ is a projective representation of the locally compact Abel group $ G $ in $ 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.