Isomorphisms of locally compact groups and Banach algebras

If G is a locally compact group, then $\mathrm{UBC}_r (G)^*$, the dual of the space of bounded right uniformly continuous complex-valued functions on G, with the Arens product is a Banach algebra. We prove in this paper a result that will have as a consequence the following: Let ${G_1},{G_2}$ be locally compact groups. Then the Banach algebras $\mathrm{UBC}_r{({G_1})^ \ast }$ and $\mathrm{UBC}_r{({G_2})^\ast}$ are isometric isomorphic if and only if ${G_1}$ and ${G_2}$ are topologically isomorphic.