My question is partially inspired by the following discussion:

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.