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  "body": "\n## Definition\n\nLet $G$ be a [[locally compact group|AHV05ZBCYW]] with a left [[Haar measure|VY0HXD3GC4]] $\\mu$ and [[modular function|63TX60RTRQ]] $\\Delta$.\nThe *group algebra* of $G$ is the space [[$L^1(G) = L^1(G, \\mu)$|SASP14MA1Z]] of integrable functions, with the following operations.\n\n1. The product is [[convolution|2Z29JN3VES]]: $(f * h)(x) = \\int_G f(y) \\, h(y^{-1}x) \\, d\\mu(y)$.\n2. The involution is $f^*(x) = \\Delta(x^{-1}) \\, \\overline{f(x^{-1})}$.\n3. The norm is $\\norm{f}_1 = \\int_G \\abs{f} \\, d\\mu$.\n\n## Remarks\n\nWith these operations $L^1(G)$ is a [[Banach algebra|9KYWN94QDN]] with an isometric involution, that is, a Banach $*$-algebra: $\\norm{f * h}_1 \\le \\norm{f}_1 \\norm{h}_1$, $(f * h)^* = h^* * f^*$ and $\\norm{f^*}_1 = \\norm{f}_1$.\n\nExample: $G = \\Z$ with counting measure.\nThen $L^1(\\Z) = \\ell^1(\\Z)$, $\\Delta \\equiv 1$, the product is $(f * h)(n) = \\sum_m f(m) h(n - m)$ and $f^*(n) = \\overline{f(-n)}$.\nThe sequence $\\delta_0$ is a unit: $(\\delta_0 * h)(n) = h(n)$.\nHere $\\ell^1(\\Z)$ contains the [[group algebra|MNP2HBWB5Q]] $\\C[\\Z]$ of finitely supported sequences as a dense subalgebra.\n\nNon-example of a unit: $L^1(\\R)$ has no unit.\nIf $e * f = f$ for all $f \\in L^1(\\R)$, then taking Fourier transforms gives $\\hat e = 1$ wherever some $\\hat f$ is nonzero, so $\\hat e \\equiv 1$, which contradicts the Riemann–Lebesgue lemma $\\hat e(\\xi) \\to 0$.\nIn general $L^1(G)$ has a unit exactly when $G$ is discrete, but it always has a bounded approximate unit.\n\n$L^1(G)$ is commutative if and only if $G$ is abelian.\n",
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