TITLE: Topology on the Unitary Dual
QUESTION [10 upvotes]: 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?
Thanks.

REPLY [3 votes]: As you pointed out in a comment, with the topology that you describe the space of $n$-dimensional representations is closed. This means that the unitary dual with that topology would be just the disjoint union of pieces, each corresponding to a dimension $n$. On the contrary, the nice aspect of the Fell topology is that it mixes all representations with all dimensions, finite and infinite.
It makes sense in the Fell topology to say that a sequence of representations of various dimensions converge to the trivial (one-dimensional) representation. In fact, it is a serious question to ask if the trivial representation is isolated in this topology or not: for discrete groups, that's the definition of Property T.
It also makes sense to say that some finite-dimensional representations converge to some infinite-dimensional ones. This happens for instance here in an elaborate context, but there are also various simple examples. Fell's topology is just a weak topology that considers all the representations as a whole.