On the irreducibility of an induced representation

A unitary representation induced from a normal subgroup of a second countable locally compact group with abelian quotient is irreducible if and only if (i) the inducing representation is irreducible with trivial stability subgroup and (ii) the restriction of the induced representation to the normal subgroup is type I. This is proved in the context of twisted group algebras using a duality result for induced representationswhich includes the Takesaki duality theorem for crossed products of von Neumann algebras (having separable pre-dual). Examples are given showing that condition (ii) above is not redundant.