My question is partially inspired by the following discussion:  

[Topology on the Unitary Dual](https://mathoverflow.net/questions/134312/topology-on-the-unitary-dual)  

Let me remind/explain how the Fell topology is defined (at least I recall the definition which I saw): let $G$ be a locally compact group. One first consider the set $Prim(G)$ of so called *primitive ideals* (those being a kernal of the irreducible representation): for $S \subset Prim(G)$ one defines $\overline{S}=\{I \in Prim(G): \bigcap S \subset I\}$. It is not too hard to show that this operation satisfies all properties of "being a closure" therefore it defines the topology on $Prim(G)$ where closed sets $S$ are those, for which $S=\overline{S}$. There is also natural mapping from the space $\hat{G}$ of all equivalence classes of irreducible representations $[\pi] \mapsto \ker \pi$ and one can define the topology on $\hat{G}$ as the weakest topology for which this map is continuous. This topology is called the *Fell topology*. According to the cited discussion, it should be true that convergence in this topology is equivalent to convergence of all matrix coefficients. However, since we deal with *classes* of representation, one should be careful how this convergence is defined: my guess would be the following. Let $([\pi\_j])\_j$ be a net in $\hat{G}$: we say that $[\pi\_j] \to [\pi]$ where:  

for each $\varepsilon >0$, compact $K \subset G$ and each $\xi\_0,\eta\_0 \in \mathcal{H}\_{\pi}$ such that $\xi\_0 \perp \eta\_0$ there exists $j$ and $\xi\_j,\eta\_j \in \mathcal{H}\_{\pi\_j}$ such that for all $x \in K$:
$$|\langle \pi\_j(x) \xi\_j,\eta\_j \rangle - \langle \pi(x) \xi\_0,\eta\_0 \rangle|<\varepsilon.$$
So my question is the following: 



> 
> **Is the convergence in the Fell topology equivalent to the convergence of matrix coefficients defined above?** 
> 
> 
> 


EDIT: I should have been more precise: there is a one-to-one correspondence between nondegenarte representations of the so called *full* $C^\*$-algebra of the group $C^\*(G)$ and unitary group representations (this correspondence preserves the notion of irreducibility). All primitive ideal are therefore in this $C^\*$-algebra. In particular my question also makes sense in the general context of $C^\*$-algebras and I'm also interested in the answer for such a question.  

EDIT 2: I corrected the obvious mistake pointed out in the answer.



Fell topology vs. convergence of matrix coefficients
In your inequality you need to evaluate the $\pi$'s somewhere! 
Unless I am mistaken unpacking [Fell (1962), Theorem 2.2 and Remark following](http://www.ams.org/mathscinet-getitem?mr=150241), $[\pi\_j]\to[\pi]$ means that for every choice of an $\varepsilon>0$, a compact set $K\subset G$, an integer $n$, and vectors $u\_1,\dots,u\_n\in\mathscr H\_\pi$, there exists a $j\_0$ such that when $j\succcurlyeq j\_0$, $\mathscr H\_{\pi\_j}$ contains $v\_1,\dots,v\_n$ such that 
$$
\left|\langle u\_i,\pi(g)u\_i\rangle-\langle v\_i,\pi\_j(g)v\_i\rangle\right|<\varepsilon
$$
for all $i\in\{1,\dots,n\}$ and all $g\in K$. In short, positive-definite functions associated with $\pi$ are *uniform-on-compacta* limits of positive-definite functions associated with the $\pi\_j$.
(See also [Dixmier (1977)](http://www.ams.org/mathscinet-getitem?mr=458185), 3.4.10 for $\mathrm C^\*$-algebras and 18.1.5 for groups.)


