Functional analysis von Neumann algebras Theorem

Murray–von Neumann classification of factors

Builds on Von Neumann algebra Commutant Partial isometry Orthogonal projection Hilbert space Algebra of bounded operators on a Hilbert space

Statement

Let $M$ be a von Neumann algebra on a separable complex Hilbert space, with commutant $M'$ . Assume that $M$ is a factor, that is, its center $M \cap M'$ is $\C I$ . For orthogonal projections $p, q \in M$ , write $p \sim q$ if there is a partial isometry $v \in M$ with $v^*v = p$ and $vv^* = q$ . Write $p \precsim q$ if $p \sim q_0$ for some projection $q_0 \in M$ with $q_0 \le q$ . Call $p$ finite if $p \sim q_0 \le p$ implies $q_0 = p$ . Then:

for all projections $p, q \in M$ , $p \precsim q$ or $q \precsim p$ ;
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 $D(p) < \infty$ if and only if $p$ is finite;
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 $\mathrm{III}$ ;
$M$ has type $\mathrm{I}_n$ or $\mathrm{I}_\infty$ if and only if $M \cong B(K)$ for a Hilbert space $K$ of dimension $n$ or $\aleph_0$ , respectively. Here $B(K)$ is the algebra of bounded operators on $K$ .

Remarks

The comparison in part 1 comes from a maximality argument. Take a maximal family of pairwise orthogonal equivalent pieces of $p$ and $q$ . It exhausts $p$ or $q$ , since otherwise, in a factor, the two remainders would contain further equivalent nonzero pieces. The 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. In type $\mathrm{II}_1$ , $D$ is the restriction of the unique faithful normal tracial state to the projections.

Examples: $M_n(\C)$ has type $\mathrm{I}_n$ , with $D(p) = \rank p$ . The hyperfinite type II1 factor has type $\mathrm{II}_1$ . The tensor product of a type $\mathrm{II}_1$ factor with $B(\ell^2)$ has type $\mathrm{II}_\infty$ . Factors of type $\mathrm{III}$ also exist; they admit no nonzero normal semifinite trace.

The 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.

Going deeper

For a student

Every von Neumann algebra on a separable Hilbert space is a direct integral of factors, so factors are the building blocks of the theory. The classification by types is the first step; the next steps distinguish factors within one type. Type $\mathrm{I}$ factors are completely understood by part 4. Type $\mathrm{III}$ factors are further divided by Connes into types $\mathrm{III}_\lambda$ , $0 \le \lambda \le 1$ , using Tomita–Takesaki theory. Type $\mathrm{II}_1$ factors carry a trace and are studied through subfactors and free probability.

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Commutant concept
Hilbert space concept
Hilbert space adjoint concept
Operator norm concept
Orthogonal projection concept
Partial isometry concept
Von Neumann algebra concept
Algebra of bounded operators on a Hilbert space object
Murray–von Neumann classification of factors statement

Tag	`QAWDC7SWKC`
URI	`m2://QAWDC7SWKC`
Kind	statement, theorem
Writing	W0 stub
MSC	46L36, 46L10
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About	Von Neumann algebra
Uses	Von Neumann algebra, Commutant, Partial isometry, Orthogonal projection, Hilbert space, Algebra of bounded operators on a Hilbert space
Changed	2 November 2026