Unitary dual of a locally compact group - Imported Knowlpedia Content

Definition

Let $G$ be a locally compact group. Its unitary dual, denoted $\widehat G$ , is the set of unitary-equivalence classes $[\pi]$ of strongly continuous irreducible unitary representations of $G$ on nonzero complex Hilbert spaces. The set is equipped with the Fell topology, defined by approximation of coefficient functions uniformly on compact subsets. This topology, not a generally available group operation, is part of the standard meaning of $\widehat G$ . For abelian $G$ , every irreducible unitary representation is one-dimensional, and $\widehat G$ recovers the Pontryagin dual group.

The Fell topology

Informally, $[\pi]$ is near a class $[\rho]$ when selected diagonal coefficients of $\rho$ can be approximated on a chosen compact subset of $G$ by finite sums of diagonal coefficients of $\pi$ . This description is independent of representatives. The resulting space need not be Hausdorff; its separation properties encode substantial representation-theoretic information Fell and Doran, Chapter VII.

Operator-algebraic interpretation

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.

Scope and nearby duals

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.

References

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.
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.