Kernel description of Fell topology - Imported Knowlpedia Content

Definition

Let $G$ be a locally compact group, let $A=C^*(G)$ be its full group $C^*$ -algebra, and identify its irreducible representations with the unitary dual $\widehat G$ . The Fell topology has the following kernel description:

[\pi]\in\overline S \quad\Longleftrightarrow\quad \bigcap_{[\rho]\in S}\ker\rho\subseteq\ker\pi

for every $S\subseteq\widehat G$ . Equivalently, its closed sets are precisely the hulls of ideals, where the hull of an ideal $I\triangleleft A$ is $\{[\pi]:I\subseteq\ker\pi\}$ . The criterion depends only on representation kernels, not on chosen representatives or realization spaces.

Relation to the primitive ideal space

Each kernel $\ker\pi$ is a primitive ideal, so the kernel map

\kappa:\widehat G\longrightarrow\operatorname{Prim}(C^*(G)), \qquad [\pi]\longmapsto\ker\pi,

is a continuous surjection onto the primitive ideal space. The displayed closure rule says precisely that the topology on $\widehat G$ is read through the hull-kernel topology on primitive ideals. This description is equivalent to Fell's coefficient-function formulation Fell, Theorem 2.2 and the group application.

The type I distinction

When $C^*(G)$ is a type I $C^*$ -algebra, $\kappa$ is bijective and identifies $\widehat G$ with $\operatorname{Prim}(C^*(G))$ . In general, inequivalent irreducible representations can share a kernel. The kernel description then cannot distinguish them: they have exactly the same neighborhoods and violate the $T_0$ separation axiom.

How to use the criterion

To prove $[\pi]\in\overline S$ , it is enough to show that every element of $C^*(G)$ annihilated by all representations in $S$ is also annihilated by $\pi$ . To separate $[\pi]$ from $\overline S$ , one seeks $a\in C^*(G)$ with $\rho(a)=0$ for every $[\rho]\in S$ but $\pi(a)\neq0$ .

References

J. M. G. Fell, “The Dual Spaces of $C^*$ -Algebras,” Transactions of the American Mathematical Society 94 (1960), 365–403. DOI record. Relevant: Theorem 2.2 and the application to group duals.
Jacques Dixmier, $C^*$ -Algebras, North-Holland, 1977. Publisher record. Relevant: primitive ideals, spectra, and the hull-kernel topology.