The Fell topology on the unitary dual

Assume the Axiom of Choice. Let $G$ be a topological group and let $\mathcal{R}$ be a set of unitary equivalence classes of strongly continuous unitary representations of $G$ containing the unitary dual $\widehat{G}$ (The unitary dual of a locally compact group). For a representation $\pi$ with class in $\mathcal{R}$, finitely many functions of positive type ${\varphi }_1,\dots ,{\varphi }_n$ associated to $\pi$ (that is, each is a single diagonal matrix coefficient ${\varphi }_i(g)=⟨\pi (g){\xi }_i,{\xi }_i⟩$ of $\pi$, Matrix coefficient of a unitary representation, Continuous positive-type functions and normalization), a compact set $Q\subseteq G$ and $ϵ>0$, put $W(\pi ;{\varphi }_1,\dots ,{\varphi }_n,Q,ϵ):=\{\rho \in \mathcal{R}:\text{ each }{\varphi }_i\text{ is within }ϵ\text{ on }Q\text{ of a finite sum of functions of positive type associated to }\rho \}\mathrm{.}$ The Fell topology on $\mathcal{R}$, and on the unitary dual $\widehat{G}\subseteq \mathcal{R}$ in particular, is the topology generated by these sets: a subset is open when it is a union of sets of this form. This is the coefficient topology of [BeH–19, §1.C]; in the notation of Weak containment of unitary representations the condition defining $W(\pi ;\varphi ,Q,ϵ)$ is the compact-uniform approximation of $\varphi$ by coefficients of $\rho$, one function at a time.

• The displayed family is a basis. Every displayed set contains its center. An empty test list gives the whole space. For a nonempty list and $\rho$ in the displayed set, choose its finite coefficient witnesses for each test; insert a zero coefficient if a witness list is empty. Thus $N={\sum }_i{n}_i>0$. Let $\delta$ be the minimum of the positive error margins $ϵ-{\sup }_Q\mathrm{\mid }{\varphi }_i-{\sum }_j{c}_{{\eta }_{i,j},{\eta }_{i,j}}\mathrm{\mid }$. The set centered at $\rho$ testing all these individual coefficients on $Q$ to accuracy $\delta \mathrm{/}N$ is contained in the original set: the sum of their new errors is strictly less than $n_i\delta \mathrm{/}N\le \delta$, which fits each error margin. For finitely many displayed sets containing $\rho$, first perform this refinement separately for each set. All resulting tests are coefficients of the SAME representation $\rho$, so their union, the finite union of compact test sets, and the minimum of their tolerances give a displayed set containing $\rho$ inside the intersection. This proves both the refinement and finite-intersection basis axioms without treating tests from unrelated centers as coefficients of one representation.
• Finite sums as tests. Allowing finite sums of diagonal coefficients as test functions generates the same topology. For a test ${\sum }_{j=1}^m{c}_{{\xi }_j,{\xi }_j}$ and error $ϵ>0$, testing its $m>0$ summands separately with error $ϵ\mathrm{/}m$ ensures that their finite-sum witnesses add to a witness for the original test. An empty sum is the zero coefficient. Conversely, every single coefficient is a one-term sum. Thus this enlargement changes the displayed basis but not the topology; it does not make every finite sum a single diagonal coefficient of the given representation.
• Hausdorffness and discreteness are not asserted. The definition guarantees only that these neighbourhoods form a topology; the companion examples page exhibits a second-countable locally compact group whose dual is not Hausdorff. Nothing here asserts that the Fell topology is discrete, and for non-compact groups it need not be.
• Choice. The Axiom of Choice is inherited from the construction of the unitary dual and from the GNS machinery used to compare coefficients; the basis verifications above use none (The Axiom of Choice).