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  "body": "\n## Definition\n\nLet $G$ be a [[locally compact group|AHV05ZBCYW]] with left Haar measure $\\mu$, and let $L^1(G)$ be its [[group algebra|6GD25S53ND]].\nFor a [[unitary representation|74BPYJNZTM]] $\\pi$ of $G$ on a Hilbert space $H$ and $f \\in L^1(G)$, let $\\pi(f)$ be the bounded operator on $H$ given by\n$$\\ip{\\pi(f)u}{v} = \\int_G f(x) \\ip{\\pi(x)u}{v} \\, d\\mu(x) \\quad (u, v \\in H).$$\nFor $f \\in L^1(G)$ put\n$$\\norm{f}_* = \\sup_\\pi \\norm{\\pi(f)},$$\nwhere $\\pi$ runs over all unitary representations of $G$.\nThe *group C\\*-algebra* $C^*(G)$ is the completion of $L^1(G)$ for the norm $\\norm{\\cdot}_*$.\nThe *reduced group C\\*-algebra* $C^*_r(G)$ is the closure of $\\set{\\lambda(f) : f \\in L^1(G)}$ in the bounded operators on $L^2(G)$, where $\\lambda$ is the left regular representation.\n\n## Remarks\n\nThe supremum is finite, since $\\norm{\\pi(f)} \\le \\norm{f}_1$, and $\\norm{\\cdot}_*$ is a norm on $L^1(G)$, since the left regular representation is injective on $L^1(G)$.\nBoth $C^*(G)$ and $C^*_r(G)$ are C*-algebras.\n\nExample: for $G = \\Z$, the Fourier transform $f \\mapsto \\sum_n f(n) z^n$ identifies $C^*(\\Z)$ and $C^*_r(\\Z)$ with the algebra $C(\\mathbb{T})$ of continuous functions on the circle with the supremum norm.\n\nNon-example: $L^1(G)$ with its own norm $\\norm{\\cdot}_1$ is in general not a C\\*-algebra.\nIn $\\ell^1(\\Z)$, let $f = \\delta_0 + \\delta_1 - \\delta_2$.\nThen $f^* * f = 3\\delta_0 - \\delta_2 - \\delta_{-2}$, so $\\norm{f^* * f}_1 = 5$, while $\\norm{f}_1^2 = 9$, and the C\\*-identity $\\norm{f^* f} = \\norm{f}^2$ fails.\n\nThere is always a surjective homomorphism $C^*(G) \\to C^*_r(G)$; it is an isomorphism exactly when $G$ is amenable.\n",
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