Duality theorem for inductive limit groups

Abstract

In this paper, we show the so-called weak duality theorem of Tannaka type for an inductive limit–type topological group $G={lim\hspace{0.17em}}_{n\to \infty }{G}_n$ in the case where each $G_n$ is a locally compact group, and $G_n$ is embedded into $G_{n+1}$ homeomorphically as a closed subgroup. First, we explain what a weak duality theorem of Tannaka type is and explain the difference between the case of locally compact groups and the case of nonlocally compact groups. Then we introduce the concept “separating system of unitary representations (SSUR),” which assures the existence of sufficiently many unitary representations. The present $G$ has an SSUR. We prove that $G$ is complete. We give semiregular representations and their extensions for $G$. Using them, we deduce a fundamental formula about “birepresentation” on $G$. Combining these results, we can prove the weak duality theorem of Tannaka type for $G$.