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  "body": "\n## Summary\n\nA subfactor is an inclusion $N \\subseteq M$ of factors, von Neumann algebras with trivial center, that share the same identity.\nFor factors of type $\\mathrm{II}_1$, Vaughan Jones attached to such an inclusion a number $[M:N]$, the index.\nIt measures the relative size of $M$ over $N$.\nThe [[Jones index theorem|31N0FH22ZT]] shows that the possible index values are surprisingly restricted.\nSubfactor theory connects operator algebras with knot invariants, quantum groups, tensor categories and conformal field theory.\n\n## Explanations\n\n### L1 human\n\nA subfactor is like a field extension or a subgroup, but for operator algebras.\nFor a subgroup $H$ of a finite group $G$, the index $[G:H]$ counts how many copies of $H$ fit into $G$.\nFor a subfactor, the index is a real number that plays the same role.\nFor example, let $N$ be a $\\mathrm{II}_1$ factor and look at the $k \\times k$ matrices with entries in $N$.\nThe diagonal matrices with all diagonal entries equal form a copy of $N$ inside them, and the index is $k^2$.\nThe surprise is that the index need not be an integer.\n",
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