A bipolar property of subgroups and translation invariant closed convex subsets

Let G be a locally compact group. We define the bipolar property of subgroups of G using the concept of the dual space G ⁎ and show that a subgroup H has the bipolar property if and only if G had H-separation property. We study generalized translation invariant closed convex subset of A ( G ) and VN ( G ) . We also prove that every completely complemented weak⁎-closed translation invariant subspace of VN ( G ) is invariantly complemented if G is amenable and give characterizations of WAP ( G ˆ ) and AP ( G ˆ ) by using the generalized translation via elements in G ⁎ .