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  "body": "\n## Statement\n\nLet $G$ be a [[locally compact group|AHV05ZBCYW]] and let $g \\in G$ with $g \\ne e$.\nThen there is an [[irreducible|FX3NEFYFFN]] [[unitary representation|74BPYJNZTM]] $\\pi$ of $G$ on a Hilbert space $H$ such that $\\pi(g) \\ne \\id_H$.\n\n## Remarks\n\nIn short, the irreducible unitary representations of a locally compact group separate its points.\nThe proof works with continuous [[positive-definite functions|3PRZ02APFA]] $\\varphi$ with $\\varphi(e) \\le 1$.\nThese form a convex set, compact in the weak-$*$ topology of $L^\\infty(G)$.\nIts nonzero extreme points are exactly the functions $x \\mapsto \\ip{\\pi(x)v}{v}$ with $\\pi$ irreducible and $v$ a unit vector.\nThe left regular representation supplies a positive-definite function that is not constant on the pair $\\set{e, g}$, and the Krein–Milman theorem then yields an extreme point with the same property.\nExample: for $G = \\R$, the characters $x \\mapsto e^{ixy}$, $y \\in \\R$, are one-dimensional irreducible unitary representations, and for $g \\ne 0$ one can choose $y$ with $e^{igy} \\ne 1$.\nFor a compact group the representations can be chosen finite-dimensional, which is part of the Peter–Weyl theorem.\n\n## Explanations\n\n### L1 human\n\nA unitary representation lets a group act by rotations on a Hilbert space, and irreducible ones are the building blocks that cannot be split further.\nThe theorem says that there are enough building blocks to see every element: each element other than the identity acts nontrivially in at least one of them.\nSo no information about the group is lost when one studies it through its irreducible unitary representations.\n\n### L2 human\n\nThe theorem justifies the program of harmonic analysis on locally compact groups, which aims to decompose functions and representations into irreducible pieces.\nIt is the starting point for the study of the unitary dual and of the group C\\*-algebra, since it shows that the irreducible representations already detect the group.\nFor abelian groups, it says that characters separate points, which is a step towards Pontryagin duality.\nIrreducible representations may need to be infinite-dimensional: for a connected simple Lie group such as $SL_2(\\R)$, every finite-dimensional unitary representation is trivial.\n",
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    "name": "Gelfand–Raikov theorem",
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  "name": "Gelfand–Raikov theorem",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/CFD/CFDZJ55N95.md",
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