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  "body": "\n## Statement\n\nLet $A$ be a commutative C*-algebra, and let $\\Delta(A)$ be its [[character space|M7JVFJ3MMX]].\nLet $C_0(\\Delta(A))$ be the C\\*-algebra of [[continuous functions vanishing at infinity|N59BW1QFD1]] on $\\Delta(A)$, with the sup norm and complex conjugation.\nThen the [[Gelfand transform|VWNSHKJT92]] $\\Gamma \\colon A \\to C_0(\\Delta(A))$, $\\Gamma(a)(\\varphi) = \\varphi(a)$, is an isometric $*$-isomorphism of $A$ onto $C_0(\\Delta(A))$.\n\n## Remarks\n\nIf $A$ has a unit, $\\Delta(A)$ is compact and $A \\cong C(\\Delta(A))$.\nProof idea: a character takes real values on self-adjoint elements, so $\\Gamma(a^*) = \\overline{\\Gamma(a)}$.\nFor self-adjoint $a$ the C\\* identity gives $\\norm{a^{2^k}} = \\norm{a}^{2^k}$, so by [[Gelfand's spectral radius formula|T4RFYT8HDN]] $\\norm{\\Gamma(a)}_\\infty = \\norm{a}$; applying this to $a^* a$ shows that $\\Gamma$ is isometric for all $a$.\nSo the range of $\\Gamma$ is a closed $*$-subalgebra that separates points and vanishes at no point, and by the [[Stone–Weierstrass theorem|1P8D3MK79H]] it is all of $C_0(\\Delta(A))$.\n\nExample: for $A = C(X)$ with $X$ compact Hausdorff, $\\Delta(A)$ is homeomorphic to $X$ through the evaluations, and $\\Gamma$ is the identity map up to this homeomorphism.\n\nIn the unital case, the theorem gives a contravariant equivalence between commutative unital C\\*-algebras with unital $*$-homomorphisms and compact Hausdorff spaces with continuous maps.\n\n## Explanations\n\n### L1 human\n\nEvery commutative C\\*-algebra is an algebra of continuous functions on some space, with nothing lost.\nThe space is recovered from the algebra as the set of its characters.\nSo studying commutative C\\*-algebras is the same as studying locally compact Hausdorff spaces.\n\n### L2 human\n\nThis theorem is the starting point of \"noncommutative topology\": a general C\\*-algebra is thought of as the functions on a space that does not exist.\nIt gives the [[continuous functional calculus|JB7S6WPZ77]].\nFor a normal element $a$, the C\\*-algebra generated by $a$ and $1$ is commutative, and its character space is the spectrum of $a$.\nSo it is isomorphic to $C(\\sigma(a))$.\nThe noncommutative Gelfand–Naimark theorem says that every C\\*-algebra is isometrically $*$-isomorphic to a closed $*$-subalgebra of the bounded operators on a Hilbert space.\n",
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