We give a characterization in
terms of Ĝ of those parts in the unitary dual of a locally compact group G, which
correspond to closed normal subgroups of G. These are exactly the sets S ⊂Ĝ,
which have the property that for all π, ρ ∈ S the support of π ⊗ρ is contained in S
and which are closed in a topology on Ĝ, which is in general weaker than the
standard topology on Ĝ, and which we call the L1-hull-kernel-topology. As
an easy consequence we obtain that for ∗-regular groups G the mapping
N → N⊥= {π ∈Ĝ|π|N= 1|ℋπ} is a bijection from the set of closed normal
subgroups of G onto the set of closed subsets S ⊂Ĝ with the property that π ⊗ρ
has support in S for all π, ρ ∈ S. This generalizes and unifies results of Pontryagin,
Helgason and Hauenschild, with a considerably simplified proof. Furthermore we
prove that ∗-regular groups have the weak Frobenius property (TP 1), i.e. 1G is
weakly contained in π ⊗π for all unitary representations π of G, generalizing a result
of E. Kaniuth.
