Tempered unitary representations

Assume the Axiom of Choice. Let $G$ be a locally compact, $\sigma$-compact group with a fixed left Haar measure, and let ${\lambda }_G$ be its left regular representation on $L^2(G)$ (Left and right regular unitary representations of an LCH group, The regular representations are unitary, strongly continuous, and the left one is faithful). A strongly continuous unitary representation $\pi$ of $G$ is tempered if it is weakly contained in ${\lambda }_G$ (Strongly continuous unitary representations, invariant linear subspaces and intertwiners, Weak containment of unitary representations); explicitly, each continuous positive-type coefficient $g\mapsto ⟨\pi (g)\xi ,\xi ⟩$ is approximable uniformly on compact subsets of $G$ by finite sums of positive-type coefficients of ${\lambda }_G$.

The reduced (tempered) dual is ${\widehat{G}}_{\mathrm{r}\mathrm{e}\mathrm{d}}:=\{[\sigma ]\in \widehat{G}:\sigma \prec {\lambda }_G\},$ the Fell support of the regular representation (The Fell topology on the unitary dual, The unitary dual of a locally compact group). Thus an irreducible unitary representation is tempered exactly when its equivalence class is a point of ${\widehat{G}}_{\mathrm{r}\mathrm{e}\mathrm{d}}$. A reducible representation may be tempered by the same weak-containment condition, but it is not itself a point of the irreducible unitary dual.

Choice. AC is inherited through the Haar-based regular representation and the set and Fell constructions of the unitary dual; the weak-containment criterion itself uses no additional choice (The Axiom of Choice).