Unitary dual

Remarks

Unitary equivalence is an equivalence relation, so $\widehat{G}$ is well defined. It is usually given the Fell topology, which comes from the identification of $\widehat{G}$ with the spectrum of the group C*-algebra $C^*(G)$.

Example: $G = \R$. By Schur's lemma, every irreducible unitary representation of an abelian group is one-dimensional, so it is a continuous homomorphism $\R \to \set{z \in \C : \abs{z} = 1}$. These are the maps $x \mapsto e^{ixy}$ with $y \in \R$, and different $y$ give inequivalent representations, so $\widehat{\R}$ is in bijection with $\R$.

Non-example: the representation of $\R$ on $L^2(\R)$ by translations $f \mapsto f(\cdot - t)$ does not define a point of $\widehat{\R}$, since it is not irreducible. The closed subspace of functions whose Fourier transform vanishes on $(-\infty, 0)$ is invariant under all translations and is neither $0$ nor $L^2(\R)$.

For a locally compact abelian group, $\widehat{G}$ is the group of continuous characters, the Pontryagin dual. For a compact group, every element of $\widehat{G}$ is finite-dimensional.