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  "body": "\n## Statement\n\nLet $M$ be a [[von Neumann algebra|PSFWQVZ0FS]] on a separable complex Hilbert space, with [[commutant|1VC7EJ1ZBZ]] $M'$.\nAssume that $M$ is a factor, that is, its center $M \\cap M'$ is $\\C I$.\nFor [[orthogonal projections|0B8KZRT28M]] $p, q \\in M$, write $p \\sim q$ if there is a [[partial isometry|GGDSF85JPD]] $v \\in M$ with $v^*v = p$ and $vv^* = q$.\nWrite $p \\precsim q$ if $p \\sim q_0$ for some projection $q_0 \\in M$ with $q_0 \\le q$.\nCall $p$ finite if $p \\sim q_0 \\le p$ implies $q_0 = p$.\nThen:\n\n1. for all projections $p, q \\in M$, $p \\precsim q$ or $q \\precsim p$;\n2. there is a dimension function $D$ from the projections of $M$ to $[0, \\infty]$. It is unique up to a positive scalar factor. It satisfies: $p \\precsim q$ if and only if $D(p) \\le D(q)$. Also $D(p + q) = D(p) + D(q)$ when $pq = 0$. And $D(p) \u003c \\infty$ if and only if $p$ is finite;\n3. after normalization, the range of $D$ is exactly one of the following sets, which defines the type of $M$. The set $\\set{0, 1, \\dots, n}$ gives type $\\mathrm{I}_n$, for $n \\ge 1$. The set $\\set{0, 1, 2, \\dots, \\infty}$ gives type $\\mathrm{I}_\\infty$. The interval $[0, 1]$ gives type $\\mathrm{II}_1$. The interval $[0, \\infty]$ gives type $\\mathrm{II}_\\infty$. The set $\\set{0, \\infty}$ gives type $\\mathrm{III}$;\n4. $M$ has type $\\mathrm{I}_n$ or $\\mathrm{I}_\\infty$ if and only if $M \\cong B(K)$ for a [[Hilbert space|HM2HHE3J51]] $K$ of dimension $n$ or $\\aleph_0$, respectively. Here $B(K)$ is the [[algebra of bounded operators|BMDG4JQPEH]] on $K$.\n\n## Remarks\n\nThe comparison in part 1 comes from a maximality argument.\nTake a maximal family of pairwise orthogonal equivalent pieces of $p$ and $q$.\nIt exhausts $p$ or $q$, since otherwise, in a factor, the two remainders would contain further equivalent nonzero pieces.\nThe dimension function $D$ is built by cutting projections into many equivalent pieces and counting, much as a length is measured with smaller and smaller rulers.\nIn type $\\mathrm{II}_1$, $D$ is the restriction of the unique faithful normal tracial state to the projections.\n\nExamples: $M_n(\\C)$ has type $\\mathrm{I}_n$, with $D(p) = \\rank p$.\nThe [[hyperfinite type II1 factor|NZ4NY8SK1C]] has type $\\mathrm{II}_1$.\nThe tensor product of a type $\\mathrm{II}_1$ factor with $B(\\ell^2)$ has type $\\mathrm{II}_\\infty$.\nFactors of type $\\mathrm{III}$ also exist; they admit no nonzero normal semifinite trace.\n\nThe hypothesis that $M$ is a factor is needed: for the non-factor $M = \\C \\oplus \\C$ acting diagonally on $\\C^2$, the two projections $(1, 0)$ and $(0, 1)$ are not comparable, so part 1 fails.\n\n## Explanations\n\n### L1 human\n\nIn a factor one can compare the \"sizes\" of projections, like comparing the dimensions of subspaces.\nMurray and von Neumann showed that this size takes values in only five kinds of ranges.\nFor matrices the sizes are whole numbers, as dimensions should be.\nIn type $\\mathrm{II}_1$ factors the sizes fill the whole interval $[0,1]$, giving \"continuous dimension\".\nIn type $\\mathrm{III}$ factors every nonzero projection is infinite.\n\n### L2 human\n\nEvery von Neumann algebra on a separable Hilbert space is a direct integral of factors, so factors are the building blocks of the theory.\nThe classification by types is the first step; the next steps distinguish factors within one type.\nType $\\mathrm{I}$ factors are completely understood by part 4.\nType $\\mathrm{III}$ factors are further divided by Connes into types $\\mathrm{III}_\\lambda$, $0 \\le \\lambda \\le 1$, using [[Tomita–Takesaki theory|NBR81PR7E5]].\nType $\\mathrm{II}_1$ factors carry a trace and are studied through subfactors and free probability.\n",
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