Double orbits of weakly almost periodic functions

A locally compact group G is called a WS-group if the double orbits of the weakly almost periodic functions on G are relatively weakly compact. It is known that Moore-groups are WS-groups. We will show that if a discrete FC-group is a WS-group then its center is of finite index in G. We will study noncompact locally compact groups with the property that if the double orbits of bounded continuous functions on G are relatively weakly compact then they are relatively norm compact. Examples of such groups include the motion group M(n) and the special linear group SL(n,R).