Duality for Sin-groups

The well-known duality theorem of Pontryagin and van Kampen [27, 20] states that the characters of an Abelian locally compact group G also form a group Ĝ, and the group of characters of Ĝ is isomorphic to G. For non-Abelian groups, there is no such duality. In a few words, difficulties on this way arise from the fact that the dual space of a noncommutative group, however defined, has no natural group structure. It was gradually realized that the statement of problem ought to be changed. An object of another type must be associated to every group, most naturally an algebra; the dual object will have the same type but may not correspond to any group. On the subclass of Abelian groups, this correspondence should respect the Pontryagin duality. This may be illustrated as follows. Let LCG and LCAG denote the categories of all and Abelian locally compact groups respectively, and ̂ the Pontryagin duality functor. One should find a category M with a duality functor * such that the following diagram is commutative, with vertical arrows being imbeddings: