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  "body": "\n## Definition\n\nLet $M$ be a [[von Neumann algebra|PSFWQVZ0FS]] on a complex [[Hilbert space|HM2HHE3J51]] $H$, with [[commutant|1VC7EJ1ZBZ]] $M'$.\nLet $\\Omega \\in H$ be cyclic for $M$, that is, $M\\Omega$ is dense in $H$.\nAssume also that $\\Omega$ is separating for $M$: if $x \\in M$ and $x\\Omega = 0$, then $x = 0$.\n\n1. The operator $S_0$ with domain $M\\Omega$ defined by $S_0(x\\Omega) = x^*\\Omega$ is a well-defined, densely defined, closable antilinear [[unbounded operator|KRJX5RA0PW]]; let $S$ be its closure.\n2. The *modular operator* is the positive self-adjoint operator $\\Delta = S^* S$, and the *modular conjugation* is the antiunitary operator $J$ in the polar decomposition $S = J \\Delta^{1/2}$.\n3. The *modular automorphism group* is $\\sigma_t(x) = \\Delta^{it} x \\Delta^{-it}$ for $t \\in \\R$ and $x \\in M$.\n\n*Tomita–Takesaki theory* is the study of these objects.\nIts main theorem states that $J M J = M'$ and $\\Delta^{it} M \\Delta^{-it} = M$ for all $t \\in \\R$.\nSo each $\\sigma_t$ is an automorphism of $M$.\n\n## Remarks\n\nExample: let $M = M_n(\\C)$ act by left multiplication on $H = M_n(\\C)$ with inner product $\\langle \\xi, \\eta \\rangle = \\operatorname{tr}(\\eta^* \\xi)$, and let $\\Omega = \\rho^{1/2}$ for an invertible density matrix $\\rho$.\nThen $\\Omega$ is cyclic and separating, $\\Delta \\xi = \\rho \\xi \\rho^{-1}$ and $J \\xi = \\xi^*$.\nCheck: $J \\Delta^{1/2}(x \\rho^{1/2}) = J(\\rho^{1/2} x) = x^* \\rho^{1/2} = S(x\\rho^{1/2})$.\nHere $JMJ$ is the algebra of right multiplications, which is $M'$, and $\\sigma_t(x) = \\rho^{it} x \\rho^{-it}$.\n\nNon-example: let $M = B(H)$ with $\\dim H \\ge 2$, let $\\Omega$ be a unit vector, and let $\\eta$ be a unit vector orthogonal to $\\Omega$.\nEvery nonzero vector is cyclic for $B(H)$, but $\\Omega$ is not separating.\nIndeed, the operator $x = \\langle \\cdot, \\eta \\rangle \\Omega$ is nonzero and $x\\Omega = 0$.\nIts adjoint is $x^* = \\langle \\cdot, \\Omega \\rangle \\eta$, so $x^*\\Omega = \\eta \\ne 0$.\nSince $x\\Omega = 0\\Omega$ but $x^*\\Omega \\ne 0^*\\Omega$, the formula for $S_0$ does not define a map.\n\nThe vector state $\\varphi(x) = \\langle x\\Omega, \\Omega \\rangle$ is a KMS state for the modular automorphism group, and Takesaki showed that $\\sigma$ is the only one-parameter automorphism group of $M$ with this property.\n",
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    "name": "Tomita–Takesaki theory",
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  "name": "Tomita–Takesaki theory",
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