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  "body": "\n## Summary\n\nThe group C*-algebra $C^*(G)$ of a [[locally compact group|AHV05ZBCYW]] $G$ is a completion of the [[group algebra|6GD25S53ND]] $L^1(G)$.\nIt uses the largest norm coming from [[unitary representations|74BPYJNZTM]] of $G$.\nIt is a C*-algebra whose nondegenerate representations correspond exactly to the unitary representations of $G$.\nThe reduced group C*-algebra $C^*_r(G)$ uses the left regular representation alone.\nThese algebras let one apply the theory of operator algebras to groups, and they are central in noncommutative geometry and the Baum–Connes conjecture.\n\n## Explanations\n\n### L1 human\n\nThe integrable functions on a group form an algebra, but its norm is not well suited to operator theory.\nOne fixes this by measuring each function through how large an operator it becomes in every unitary representation, and then filling in the gaps.\nFor the integers, this turns absolutely summable sequences into all continuous functions on the circle, via Fourier series.\nSo $C^*(\\Z)$ is the algebra $C(\\mathbb{T})$ of continuous functions on the circle.\n",
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