A strong duality theorem for separable locally compact groups

Abstract:

We obtain a duality theorem for separable locally compact groups, where the group is regained from the set of factor unitary representations. Loosely stated, the group is isomorphic to the group of nonzero bounded, operator valued maps on the set of factor representations, which preserve unitary equivalence, direct sums, and tensor products. The axiom involving tensor products is formulated in terms of direct integral theory. The topology of G may be regained from the irreducible representations alone. Indeed a sequence $\{ {x_i}\}$ in G, converges to x in G if and and only if $\pi ({x_i})$ converges strongly to $\pi (x)$ for each irreducible representation $\pi$ of G. This result supplies the missing topological part of the strong duality theorem of N. Tatsuuma for type I separable locally compact groups (based on irreducible representations). Our result also generalizes this Tatsuuma strong duality theorem to the nontype I case.