The weak closure of the set of left translation operators

It is known that for an amenable locally compact group G, 0 is not in the weak closure of {lambda(g) : g is an element of G}of VN(G). In this paper, it is proved that the converse of this is true. In other words, if G is a non-amenable locally compact group, then 0 is in the weak closure of {lambda(g) : g is an element of G}. This answers several questions of Ulger. Applications to the algebra C-delta*(G) and the dual of the reduced group C*-algebra are obtained.