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  "body": "\n## Definition\n\nLet $H$ be a Hilbert space and let $B(H)$ be the [[algebra of bounded operators|BMDG4JQPEH]] on $H$.\nFor a set $\\mathcal{S} \\subseteq B(H)$, let $\\operatorname{Lat} \\mathcal{S}$ be the set of closed subspaces of $H$ that are [[invariant|0ACX5QT8TE]] under every operator in $\\mathcal{S}$.\nFor a set $\\mathcal{L}$ of closed subspaces of $H$, let\n$$\\operatorname{Alg} \\mathcal{L} = \\set{T \\in B(H) : T(L) \\subseteq L \\text{ for all } L \\in \\mathcal{L}}.$$\nA subalgebra $\\mathcal{A} \\subseteq B(H)$ is *reflexive* if $\\mathcal{A} = \\operatorname{Alg} \\operatorname{Lat} \\mathcal{A}$.\n\n## Remarks\n\nAlways $\\mathcal{A} \\subseteq \\operatorname{Alg} \\operatorname{Lat} \\mathcal{A}$, so reflexivity is the reverse inclusion.\nA reflexive algebra contains the identity and is closed in the weak operator topology, because each $\\operatorname{Alg} \\mathcal{L}$ has these properties.\nExample: $\\mathcal{A} = B(H)$ is reflexive, since $\\operatorname{Lat} B(H) = \\set{\\set{0}, H}$ and $\\operatorname{Alg} \\set{\\set{0}, H} = B(H)$.\nNon-example: in $M_2(\\C)$, let $\\mathcal{A}$ be the matrices $\\begin{pmatrix} a \u0026 b \\\\ 0 \u0026 a \\end{pmatrix}$.\nIts invariant subspaces are those of $\\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$, namely $\\set{0}$, $\\C e_1$ and $\\C^2$, so $\\operatorname{Alg} \\operatorname{Lat} \\mathcal{A}$ is the algebra of all upper triangular matrices, which is strictly larger.\nEvery von Neumann algebra is reflexive: its invariant subspaces are the ranges of the projections in its commutant, and the bicommutant theorem gives the equality.\n",
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