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  "body": "\n## Summary\n\nThe ultraweak topology, also called the $\\sigma$-weak topology, is a topology on the bounded operators $B(H)$ on a Hilbert space $H$.\nIt is the weak-$*$ topology that comes from viewing $B(H)$ as the dual of the space of [[trace class operators|Z2CP530WDE]].\nIt is finer than the [[weak operator topology|4XM1VN1C1N]] but agrees with it on bounded sets.\nThe linear functionals that are continuous for it are exactly the normal functionals, which makes it the intrinsic topology of a [[von Neumann algebra|PSFWQVZ0FS]].\n\n## Explanations\n\n### L1 human\n\nThe weak operator topology tests an operator $T$ against finitely many pairs of vectors at a time.\nThe ultraweak topology allows infinitely many pairs at once, as long as the total size of the vectors is finite.\nThis is the same as testing $T$ against density matrices $\\rho$ through the number $\\operatorname{tr}(T \\rho)$, the expected value of $T$ in the state $\\rho$ in quantum mechanics.\nOn the unit ball nothing changes, but globally the topology has more open sets.\nIts advantage is that it depends only on the algebra and not on the Hilbert space on which it acts.\n",
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