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  "body": "\n## Definition\n\nLet $G$ be a [[locally compact group|AHV05ZBCYW]].\nTwo [[unitary representations|74BPYJNZTM]] $\\pi_1$ on $H_1$ and $\\pi_2$ on $H_2$ are *unitarily equivalent* if there is a unitary operator $T \\colon H_1 \\to H_2$ with $T\\pi_1(g) = \\pi_2(g)T$ for all $g \\in G$.\nThe *unitary dual* $\\widehat{G}$ of $G$ is the set of unitary equivalence classes of [[irreducible|FX3NEFYFFN]] unitary representations of $G$.\n\n## Remarks\n\nUnitary equivalence is an equivalence relation, so $\\widehat{G}$ is well defined.\nIt is usually given the Fell topology, which comes from the identification of $\\widehat{G}$ with the spectrum of the group C\\*-algebra $C^*(G)$.\n\nExample: $G = \\R$.\nBy Schur's lemma, every irreducible unitary representation of an abelian group is one-dimensional, so it is a continuous homomorphism $\\R \\to \\set{z \\in \\C : \\abs{z} = 1}$.\nThese are the maps $x \\mapsto e^{ixy}$ with $y \\in \\R$, and different $y$ give inequivalent representations, so $\\widehat{\\R}$ is in bijection with $\\R$.\n\nNon-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.\nThe closed subspace of functions whose Fourier transform vanishes on $(-\\infty, 0)$ is invariant under all translations and is neither $0$ nor $L^2(\\R)$.\n\nFor a locally compact abelian group, $\\widehat{G}$ is the group of continuous characters, the Pontryagin dual.\nFor a compact group, every element of $\\widehat{G}$ is finite-dimensional.\n",
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