Suppose I have a locally compact topological group G. The unitary dual of G is the set of equivalence classes of irreducible unitary representations of G. Now, it seems to me that the sensible way of putting a topology on this space is as follows:
Fix a hilbert space Hn of cardinality n.
Consider the set R(G,Hn), the set of unitary representations of G on Hn. We can give it the topology of uniform convergence on compact sets. Specifically, reps pn approach p if for any compact K in G and v in Hn, pn(g)v -> p(g)v uniformly on K.
Now take the subspace I(G,Hn) of irreducible representations, with the subspace topology. Then quotient by unitary equivalence, and give the resulting space the quotient topology.
Finally, take a disjoint union over all (countable) n.
I am not sure, however, if this is commonly done. The popular topology on the unitary dual seems to be the Fell topology. Is what I described equivalent? If not, what advantages does the Fell topology have? Also, there is the perspective that the unitary dual is more importantly a measure space than a topological space- is a topological structure significant or important?