Unitary dual of a locally compact group

For abelian $G$, every irreducible unitary representation is one-dimensional, and $\widehat G$ recovers the Pontryagin dual group.

For a neighborhood centered at $[\pi]$, the approximation condition asks that selected diagonal coefficients of $\pi$ be approximated on a chosen compact subset of $G$ by finite sums of diagonal coefficients of the variable representation $\rho$. This description is independent of representatives. The resulting space need not be Hausdorff; its separation properties encode substantial representation-theoretic information.

Strongly continuous unitary representations of $G$ correspond to nondegenerate representations of the full group $C^*$-algebra $C^*(G)$. Consequently, $\widehat G$ is naturally the spectrum of irreducible representations of $C^*(G)$. Passing from a representation to its kernel maps this spectrum onto the primitive ideal space, but distinct irreducible representations can have the same kernel unless additional hypotheses, such as type I regularity, are imposed.

The unitary dual includes all irreducible unitary representations. The tempered dual is generally a proper subspace defined by weak containment in the regular representation. Nor should $\widehat G$ be confused with the algebraic dual of a Lie algebra or with the set of all continuous characters; these coincide with the relevant unitary dual only in special abelian settings.

1. J. M. G. Fell and R. S. Doran, Representations of $*$-Algebras, Locally Compact Groups, and Banach $*$-Algebraic Bundles, vol. 1, Academic Press, 1988. Publisher record. Relevant: Chapter VII on representation spaces and their topology.
2. Jacques Dixmier, $C^*$-Algebras, North-Holland Mathematical Library 15, North-Holland, 1977. Publisher record. Relevant: chapters on spectra and the dual of a locally compact group.