: In this thesis, for a locally compact group G , we introduce and study the C ? -algebra GL 0( G ) of all generalised functions on G that vanish at infinity. Among the other results, we give a result about the structure of GL 0( G ) as a Banach suace of M ( G ) ? ; in fact, we show that GL _( G ) is not a complemented suace of GL ( G ). Also, we deal with the second dual of the measure algebra M ( G ) for a locally compact group G under certain locally convex topologies. We first introduce and study a locally convex topology ? ( G ) on M ( G ) under which the Banach space GL _( G ) can be identified with the strong dual of M ( G ). Here, an application of this result is made to define and investigate an Arens multiplication on the second dual of ( M ( G ) , ? ( G )) ?? . As the main result, we show that if G ?? and G _ are locally compact groups and T is an isometric isomorphism from the Banach algebra ( M ( G 1) , ? ( G 1)) ?? onto ( M ( G 2) , ? ( G 2)) ?? , then G 1 and G 2 are homeomorphic.