Fell topology

Remarks

The operation $S \mapsto \overline{S}$ is a closure operator, so this defines a topology; it is the pull-back of the Jacobson topology on the primitive ideals of $C^*(G)$. Fell showed an equivalent form: $\pi \in \overline{S}$ exactly when $\pi$ is weakly contained in $S$. This means that every function $x \mapsto \ip{\pi(x)\xi}{\xi}$ is a limit, uniformly on compact subsets of $G$, of finite sums of functions $x \mapsto \ip{\sigma(x)\eta}{\eta}$ with $\sigma \in S$.

Example: let $G = \Z$, so $\widehat{G}$ is the set of characters $\chi_\theta(n) = e^{in\theta}$, $\theta \in \R/2\pi\Z$. If $\theta_k \to \theta$, then $\chi_{\theta_k} \to \chi_\theta$ uniformly on finite sets, so $\chi_\theta$ lies in the closure of $\set{\chi_{\theta_k}}$. In general, for an abelian group the Fell topology is the compact-open topology of the Pontryagin dual.

Non-example: in $\widehat{\Z}$, the trivial character $\chi_0$ is not in the closure of $\set{\chi_\pi}$, where $\chi_\pi(n) = (-1)^n$. The positive-type functions of $\chi_\pi$ and their sums are the functions $c(-1)^n$ with $c \ge 0$. To approach $\chi_0$ one would need $c \to 1$ at $n = 0$ and $-c \to 1$ at $n = 1$, which is impossible.

For a compact group the Fell topology is discrete. The Fell topology is $T_1$ for many groups but not always Hausdorff.