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  "body": "\n## Summary\n\nA von Neumann algebra is an algebra of bounded operators on a Hilbert space that contains the identity and is closed under taking adjoints.\nIt must also be closed in the [[weak operator topology|4XM1VN1C1N]].\nIt is a special kind of C*-algebra, one that is rich in projections and admits a measurable functional calculus.\nCommutative von Neumann algebras are the algebras $L^\\infty$ of measure spaces, so the theory is often called noncommutative measure theory.\nJohn von Neumann introduced them and studied them with Francis Murray.\nThey are basic objects in operator algebras, quantum physics and representation theory.\n\n## Explanations\n\n### L1 human\n\nTake some operators on a Hilbert space and keep adding sums, products, adjoints and limits.\nIf the limits are taken in the norm, one gets a C*-algebra.\nIf they are taken in the much weaker sense of matrix coefficients, one gets a von Neumann algebra.\nThe weaker limits add many projections, the operator analogues of indicator functions of sets.\nFor example, the continuous functions on $[0,1]$, acting by multiplication on $L^2[0,1]$, do not form a von Neumann algebra.\nAdding the weak limits gives all bounded measurable functions, which do.\nThe matrix algebras $M_n(\\C)$ are the simplest examples.\n",
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