The relation between existence of admissible vectors and compactness of a locally compact group

Let G be a locally compact group and : G ?! U(H ) be a unitary representation. In this article we will study on the existence of an admissible vector for irreducible representations and the compactness of G. In fact, it will be shown that if G is a compact group then every irreducible representation of G has an admissible vector, and also has a bounded cyclic vector. Conversely if G has property (T) and a finite irreducible representation of G has an admissible vector, then G is a compact group. Since containment of an irreducible representation in the left regular representation G is a necessary and su cient condition for existence of admissible vector, hence it seems a natural object of weakly containment of these representation and therefore property (T) is peered.

Keywords:

admissible vector , property (T) , compact group.