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  "body": "\n## Definition\n\nLet $A$ be a C*-algebra.\nFor a C\\*-algebra $B$, let $A \\odot B$ be the algebraic [[tensor product|1735VGRT2P]] of $A$ and $B$, a $*$-algebra with $(a \\otimes b)^* = a^* \\otimes b^*$.\nA *C\\*-norm* on $A \\odot B$ is a norm $\\gamma$ with $\\gamma(xy) \\le \\gamma(x) \\gamma(y)$, $\\gamma(x^*) = \\gamma(x)$ and $\\gamma(x^* x) = \\gamma(x)^2$.\nThe algebra $A$ is *nuclear* if, for every C\\*-algebra $B$, there is exactly one C\\*-norm on $A \\odot B$.\n\n## Remarks\n\nThere is always a smallest C\\*-norm $\\norm{\\cdot}_{\\min}$ and a largest $\\norm{\\cdot}_{\\max}$, so $A$ is nuclear when $\\norm{\\cdot}_{\\min} = \\norm{\\cdot}_{\\max}$ on $A \\odot B$ for all $B$.\nBy theorems of Choi–Effros and Kirchberg, $A$ is nuclear if and only if it has the completely positive approximation property.\nThis means that there are nets of [[completely positive|CGY299QMC3]] contractions $\\varphi_\\lambda \\colon A \\to M_{n_\\lambda}(\\C)$ and $\\psi_\\lambda \\colon M_{n_\\lambda}(\\C) \\to A$ with $\\psi_\\lambda(\\varphi_\\lambda(a)) \\to a$ in norm for every $a \\in A$.\nA C\\*-algebra is nuclear if and only if it is an [[amenable Banach algebra|6P9AE21Q7X]].\n\nExample: $M_n(\\C)$ is nuclear.\nTake $\\varphi_\\lambda = \\psi_\\lambda$ equal to the identity map of $M_n(\\C)$, which is completely positive and contractive.\nThen $\\psi_\\lambda(\\varphi_\\lambda(a)) = a$, so the completely positive approximation property holds.\nCommutative C\\*-algebras and [[AF algebras|4Y7VENNBVK]] are also nuclear.\n\nNon-example: let $\\mathbb{F}_2$ be the free group on two generators.\nFor a discrete group $G$, the reduced group C*-algebra $C^*_r(G)$ is nuclear if and only if $G$ is amenable; since $\\mathbb{F}_2$ is not amenable, $C^*_r(\\mathbb{F}_2)$ is not nuclear.\nThe algebra $B(H)$ of all bounded operators on an infinite-dimensional Hilbert space is not nuclear either.\n",
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