Geometric properties of some subspaces of 𝑉𝑁(𝐺)

Let G be a locally compact group. We show that any one of the spaces U C B ( G ^ ) , W A P ( G ^ ) , A P ( G ^ ) UCB(\hat G),WAP(\hat G),AP(\hat G) , and C δ ∗ ( G ) C_\delta ^ \ast (G) is Asplund if and only if the group G is finite. We also show that any one of the spaces V N ( G ) , U C B ( G ^ ) VN(G),UCB(\hat G) , and C δ ∗ ( G ) C_\delta ^\ast (G) has the DPP if and only if the group G has an abelian subgroup of finite index.