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  "body": "\n## Statement\n\nLet $H$ be a complex [[Hilbert space|HM2HHE3J51]] with [[algebra of bounded operators|BMDG4JQPEH]] $B(H)$.\nLet $M$ be a subalgebra of $B(H)$ that contains the identity $I$ and is closed under the [[adjoint|6HSJBJ8TTQ]].\nLet $M''$ be the [[bicommutant|1VC7EJ1ZBZ]] of $M$.\nThen the following are equivalent:\n\n1. $M = M''$;\n2. $M$ is closed in the [[weak operator topology|4XM1VN1C1N]];\n3. $M$ is closed in the [[strong operator topology|KHJ2G2R5G7]].\n\nIn particular, $M''$ is the closure of $M$ in each of these two topologies.\n\n## Remarks\n\nEvery commutant is weakly closed, so 1 implies 2, and 2 implies 3 because the strong topology is finer.\nFor 3 implies 1, let $T \\in M''$ and $x \\in H$.\nThe [[orthogonal projection|0B8KZRT28M]] $P$ onto the closure of $Mx$ commutes with $M$ because $M$ is closed under adjoints, so $T$ commutes with $P$ and $Tx \\in \\overline{Mx}$, using $x = Ix \\in Mx$.\nApplying this to the diagonal action of $M$ on $H^n$ handles $n$ vectors at once, which shows that $T$ is a strong limit of elements of $M$.\n\nThe identity is needed: the algebra $M = \\set{0}$ is closed under adjoints and weakly closed, but $M' = B(H)$ and $M'' = \\C I \\ne M$.\nMore generally, without the identity the theorem holds in the form $M'' = \\overline{M}$ only when $M$ acts nondegenerately.\n\nFor example, on $\\C^n$ the diagonal matrices $D$ satisfy $D' = D$, so $D'' = D' = D$, as the theorem predicts for a finite-dimensional, hence closed, algebra.\n\n## Explanations\n\n### L1 human\n\nA von Neumann algebra can be described in two very different ways.\nOne is analytic: it is closed under weak limits of operators.\nThe other is algebraic: it consists of all operators that commute with everything that commutes with it.\nThe theorem says these two descriptions agree, as long as the algebra contains the identity and is closed under adjoints.\n\n### L2 human\n\nThe theorem lets one produce von Neumann algebras for free.\nThe commutant of any self-adjoint set $S$ of operators is one, and so is the bicommutant $S''$, the von Neumann algebra generated by $S$.\nIt turns questions about topological closure into algebraic questions about commuting operators.\nThis is how one shows, for example, that the spectral projections of a self-adjoint element of $M$ lie in $M$.\nTogether with the Kaplansky density theorem, it is the basic tool for approximating elements of $M''$ by elements of $M$.\n",
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