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  "body": "\n## Definition\n\nLet $A$ be a C*-algebra with unit $1$, and let $a \\in A$ be normal, that is, $a^* a = a a^*$.\nLet $\\sigma(a) \\subseteq \\C$ be the [[spectrum|EH0CZ19RKP]] of $a$, and let $C(\\sigma(a))$ be the C\\*-algebra of [[continuous|0X3WGX95FE]] functions $\\sigma(a) \\to \\C$ with the sup norm.\nThere is a unique $*$-homomorphism $\\Phi_a \\colon C(\\sigma(a)) \\to A$ with $\\Phi_a(1) = 1$ and $\\Phi_a(\\iota) = a$, where $\\iota(z) = z$ is the inclusion of $\\sigma(a)$ into $\\C$.\nThe map $\\Phi_a$ is the *continuous functional calculus* of $a$, and for $f \\in C(\\sigma(a))$ one writes $f(a) = \\Phi_a(f)$.\n\n## Remarks\n\nExistence comes from the [[commutative Gelfand–Naimark theorem|NT564Y315M]] applied to the commutative C\\*-algebra $C^*(a, 1)$ generated by $a$ and $1$, whose character space is homeomorphic to $\\sigma(a)$.\nUniqueness holds because the polynomials in $z$ and $\\overline{z}$ are dense in $C(\\sigma(a))$, by the Stone–Weierstrass theorem.\nThe map $\\Phi_a$ is isometric, its image is $C^*(a, 1)$, and $\\sigma(f(a)) = f(\\sigma(a))$.\n\nExample: $A = M_n(\\C)$ and $a = U D U^*$ with $U$ unitary and $D = \\operatorname{diag}(\\lambda_1, \\ldots, \\lambda_n)$.\nHere $\\sigma(a) = \\set{\\lambda_1, \\ldots, \\lambda_n}$ and $f(a) = U \\operatorname{diag}(f(\\lambda_1), \\ldots, f(\\lambda_n)) U^*$.\nThis map is linear, multiplicative, preserves adjoints, sends $1$ to $I$ and $\\iota$ to $a$.\n\nNon-example: $a = \\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$ in $M_2(\\C)$ is not normal.\nIts spectrum is $\\set{0}$, so $\\iota$ is the zero function on $\\sigma(a)$, and any linear map sends it to $0 \\ne a$.\nSo no such $\\Phi_a$ exists, and normality cannot be dropped.\n",
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