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  "body": "\n## Description\n\nThe hyperfinite type $\\mathrm{II}_1$ factor $R$ is the [[von Neumann algebra|PSFWQVZ0FS]] obtained as the closure of an increasing union of matrix algebras.\nEmbed $M_{2^n}(\\C)$ into $M_{2^{n+1}}(\\C)$ by $x \\mapsto I_2 \\otimes x = \\begin{pmatrix} x \u0026 0 \\\\ 0 \u0026 x \\end{pmatrix}$.\nThese embeddings preserve the normalized trace $\\tau(x) = 2^{-n} \\operatorname{tr}(x)$, so $\\tau$ is well defined on the union $A_\\infty = \\bigcup_n M_{2^n}(\\C)$.\nLet $H$ be the [[Hilbert space|HM2HHE3J51]] completion of $A_\\infty$ for the inner product $\\langle x, y \\rangle = \\tau(y^* x)$, and let $A_\\infty$ act on $H$ by left multiplication.\nThen $R$ is the closure of $A_\\infty$ in the [[weak operator topology|4XM1VN1C1N]] on $H$.\n\n## Remarks\n\nThe algebra $R$ is a factor: its center consists of scalars.\nThe trace $\\tau$ extends to a faithful normal tracial state on $R$, and the values of $\\tau$ on projections fill $[0, 1]$, so $R$ has type $\\mathrm{II}_1$.\nIt is hyperfinite: it contains an increasing sequence of finite-dimensional subalgebras whose union is weakly dense.\nMurray and von Neumann proved that a hyperfinite $\\mathrm{II}_1$ factor with separable predual is unique up to isomorphism.\nConnes proved that, among $\\mathrm{II}_1$ factors with separable predual, $R$ is also the unique injective one.\nOther constructions give the same factor, for example the group von Neumann algebra of the group of finitely supported permutations of $\\N$.\nEvery type $\\mathrm{II}_1$ factor contains a von Neumann subalgebra isomorphic to $R$.\n\n## Explanations\n\n### L1 human\n\nMatrices of size $2 \\times 2$, $4 \\times 4$, $8 \\times 8$, and so on, sit inside each other by repeating a matrix twice along the diagonal.\nMeasured with the normalized trace, which gives the identity size $1$, the projections in these algebras have sizes $k / 2^n$.\nTaking a suitable limit of all of them gives the factor $R$, where projections can have any size between $0$ and $1$.\nIt is the simplest and most important example of an operator algebra with this \"continuous dimension\".\n",
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