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  "body": "\n## Summary\n\nLet $S$ be a set of operators on a Hilbert space.\nIts commutant $S'$ is the set of all bounded operators that commute with every element of $S$.\nIt is the operator version of the centralizer in an algebra.\nThe commutant of a self-adjoint set is always a [[von Neumann algebra|PSFWQVZ0FS]].\nBy the [[bicommutant theorem|BSA6GFT8SQ]], the von Neumann algebras are the self-adjoint unital algebras equal to their double commutant.\n\n## Explanations\n\n### L1 human\n\nGiven some operators, ask which operators can be applied before or after them with the same result.\nThese form the commutant.\nFor example, the only $n \\times n$ matrices that commute with all $n \\times n$ matrices are the multiples of the identity.\nIn the other direction, the matrices that commute with all diagonal matrices are exactly the diagonal matrices.\nSo a big set has a small commutant, and a small set has a big commutant.\n",
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