UNIVERSAL APPROACH TO THE TAKESAKI-TAKAI γ-DUALITY

In this article, we generalize and simplify the proof of the TakesakiTakai 7-duality theorem. Assume a morphism omega : G Ant (A) is a projective representation of the locally compact Abel group G in Ant (A), mapping 7 : G G is continuous, and (A, , G, omega ) is a dynamic system then there exists isomorphism gamma (L 1 ( Upsilon : Env omega A circle times LK (L2 2 ( G ) ) G, , Env omega gamma(L1 omega gamma (L 1 ( G, A) ))) which is the equivariant for the double dual action (gamma gamma ( omega : G Ant L1 1 (gamma gamma (L1 1 ( G, A) )))) Env omega G, omega G , Env omega omega . These results deepen our understanding of the representation theory and are especially interesting given their possible applications to problems of the quantum theory.