Duality theorems and topological structures of groups

Abstract

We introduce four different notions of weak Tannaka-type duality theorems, and we define three categories of topological groups, called T-type groups, strongly T-type groups, and NOS-groups.

We call a one-parameter subgroup a nontrivial homomorphic image of the additive group $\mathbf{R}$ of real numbers into a topological group $G$. When $G$ does not contain any one-parameter subgroup, we call $G$ a NOS-group.

The aim of this paper is to show the following relations. In the table below, the symbol $⟺$ means that for a given topological group $G$ the duality theorem on the left-hand side holds if and only if $G$ is of type cited on the right-hand side:

(1) u-duality $⟺$ T-type,

(2) i-duality $⟺$ strongly T-type,

(3) b-duality $⟺$ locally compact,

(4) c-duality $⟺$ locally compact NOS.

We give in the last section some examples which show the actual differences among (1)–(4).