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  "body": "\n## Statement\n\nLet $G$ be a second countable, [[unimodular|CXG6JZGMCK]] [[locally compact group|AHV05ZBCYW]] of type I, with a fixed [[Haar measure|VY0HXD3GC4]] $dg$.\nLet $\\widehat{G}$ be its [[unitary dual|95Q8WZMN0W]], the set of equivalence classes of irreducible [[unitary representations|74BPYJNZTM]] of $G$, with its Borel structure.\nFor $f \\in L^1(G)$ and a unitary representation $\\pi$ of $G$ on a [[Hilbert space|HM2HHE3J51]] $\\mathcal{H}_\\pi$, let $\\pi(f) = \\int_G f(g) \\pi(g) \\, dg$, a bounded operator on $\\mathcal{H}_\\pi$.\nThen there is a unique positive Borel measure $\\mu$ on $\\widehat{G}$, the *Plancherel measure*, with the following property.\nFor every $f \\in L^1(G) \\cap L^2(G)$, the operator $\\pi(f)$ is Hilbert–Schmidt for $\\mu$-almost every $\\pi$, and\n$$\\int_G \\abs{f(g)}^2 \\, dg = \\int_{\\widehat{G}} \\norm{\\pi(f)}_{\\mathrm{HS}}^2 \\, d\\mu(\\pi).$$\n\n## Remarks\n\nThe idea of the proof is to decompose the two-sided regular representation of $G \\times G$ on $L^2(G)$.\nFor a type I group, the left regular representation is a direct integral of irreducible representations over $\\widehat{G}$.\nThe Plancherel measure is the measure in this direct integral, normalised by the choice of $dg$.\nThe identity then extends by continuity to a unitary isomorphism from $L^2(G)$ onto a direct integral of the spaces of Hilbert–Schmidt operators on the $\\mathcal{H}_\\pi$.\n\nFor $G = \\R$ with Lebesgue measure, $\\widehat{G}$ is the set of characters $x \\mapsto e^{i \\xi x}$ with $\\xi \\in \\R$, each $\\pi(f)$ is multiplication by a value of the Fourier transform, and $\\mu = d\\xi / 2\\pi$.\nThis is the classical Plancherel theorem.\nFor a [[compact group|EVHCE5VR6Y]] with Haar measure of total mass $1$, the dual $\\widehat{G}$ is discrete, $\\mu(\\set{\\pi}) = \\dim \\mathcal{H}_\\pi$, and the formula is the Parseval identity of the [[Peter–Weyl theorem|6VC9QE39XH]].\n\nThe type I hypothesis cannot simply be dropped.\nFor groups that are not of type I, the regular representation can decompose in essentially different ways, and the formula needs factor representations.\nWithout unimodularity, the operators $\\pi(f)$ must be corrected by unbounded positive operators before the formula holds.\n\n## Explanations\n\n### L1 human\n\nFor functions on the line, the Plancherel theorem says that the energy of a function equals the energy of its Fourier transform.\nFor a general group, the role of the frequencies is played by the irreducible representations of the group.\nThe theorem gives a measure on these representations, the Plancherel measure, that makes the energy identity true.\nIt tells which representations really occur when one breaks $L^2(G)$ into pieces.\n\n### L2 human\n\nMany groups met in practice are of type I: abelian groups, compact groups, connected semisimple Lie groups and connected nilpotent Lie groups.\nFor abelian groups the Plancherel measure is the Haar measure on the Pontryagin dual, suitably normalised.\nFor a semisimple Lie group, Harish-Chandra computed the Plancherel measure explicitly; the discrete series representations are exactly the atoms of $\\mu$.\nFor a [[Gelfand pair|W38ZDXDVYC]] $(G, K)$ and functions that are bi-invariant under $K$, the theorem becomes the Plancherel theorem for [[spherical functions|BKZNC76RC8]].\n",
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