C~*-algebra On Locally Compact Amenable Group

Let G be a locally compact group with C r*(G ) its reduced C *-algebra . Firstly, this article introduces the definition and properties of locally compact amenable group .Secondly, we summary the properties of C *-algebras on amenable groupG ,and we discuss the relationship between the amenable groupG and C r*(G ),then we mainly show that C r*(G ) is residually finite dimensional if and only if G is both amenable and maximally almost periodic. At last, we give an example that the discrete group is amenable .