Duality theorems and topological structures of groups

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 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 double left right arrow 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 double left right arrow T-type, (2) i-duality double left right arrow strongly T-type, (3) b-duality double left right arrow locally compact, (4) c-duality double left right arrow locally compact NOS. We give in the last section some examples which show the actual differences among (1)-(4).