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> Question: Can the Fell topology be expressed in terms of the distributions of the the tracial states of a unitary representations, that, is $\pi\_j \rightarrow \pi$ if and only if $tr\; \pi\_j \rightarrow tr\;\pi$?
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This makes of course only sense in a more restricted setting, say reductive groups over a local field $F$. Otherwise, it isn't clear whether irreducible unitary reps have tracial states.


Examples: This is true for $GL(n,F\_v)$ for $F\_v$ local field and $n=1,2$. Also works well for $SL(2, \mathbb{R})$.



Fell topology in terms of distributions
Here is a result by D. Milicic, On $C^{\ast} $-algebras with bounded trace, Glasnik Mat. Ser. III 8(28) (1973), 7–22. Say that a $C^\*$-algebra $A$ has bounded trace if the linear span of $T(A^+)$ is dense in $A$, where $T(A^+)$ is the set of those positive elements such that $\pi\mapsto Tr\,\pi(x)$ is bounded on $\hat{A}$. Milicic proves that, for such a $C^\*$-algebra $A$, a set $S\subset\hat{A}$ is the set of limits of a net $(\pi\_j)$ in $\hat{A}$, if and only if there exists positive integers $n\_\sigma$ (for $\sigma\in S$) such that, in your sense, $Tr\,\pi\_j$ converges to $\sum\_{\sigma\in S} n\_\sigma.Tr\,\sigma$. 


A result of Fell tells you that the reduced $C^\*$-algebra of a semi-simple Lie group with finite centre, has bounded trace.


