Finite von Neumann algebra

Let ${\displaystyle {\mathcal {M}}}$ denote a finite von Neumann algebra with center ${\displaystyle {\mathcal {Z}}}$. One of the fundamental characterizing properties of finite von Neumann algebras is the existence of a center-valued trace. A von Neumann algebra ${\displaystyle {\mathcal {M}}}$ is finite if and only if there exists a normal positive bounded map ${\displaystyle \tau :{\mathcal {M}}\to {\mathcal {Z}}}$ with the properties:

• ${\displaystyle \tau (AB)=\tau (BA),A,B\in {\mathcal {M}}}$,
• if ${\displaystyle A\geq 0}$ and ${\displaystyle \tau (A)=0}$ then ${\displaystyle A=0}$,
• ${\displaystyle \tau (C)=C}$ for ${\displaystyle C\in {\mathcal {Z}}}$,
• ${\displaystyle \tau (CA)=C\tau (A)}$ for ${\displaystyle A\in {\mathcal {M}}}$ and ${\displaystyle C\in {\mathcal {Z}}}$.

The finite-dimensional von Neumann algebras can be characterized using Wedderburn's theory of semisimple algebras. Let C^{n × n} be the n × n matrices with complex entries. A von Neumann algebra M is a self adjoint subalgebra in C^{n × n} such that M contains the identity operator I in C^{n × n}.

Every such M as defined above is a semisimple algebra, i.e. it contains no nilpotent ideals. Suppose M ≠ 0 lies in a nilpotent ideal of M. Since M* ∈ M by assumption, we have M*M, a positive semidefinite matrix, lies in that nilpotent ideal. This implies (M*M)^k = 0 for some k. So M*M = 0, i.e. M = 0.

The center of a von Neumann algebra M will be denoted by Z(M). Since M is self-adjoint, Z(M) is itself a (commutative) von Neumann algebra. A von Neumann algebra N is called a factor if Z(N) is one-dimensional, that is, Z(N) consists of multiples of the identity I.

Theorem Every finite-dimensional von Neumann algebra M is a direct sum of m factors, where m is the dimension of Z(M).

Proof: By Wedderburn's theory of semisimple algebras, Z(M) contains a finite orthogonal set of idempotents (projections) {P_i} such that P_iP_j = 0 for i ≠ j, Σ P_i = I, and

${\displaystyle Z(\mathbf {M} )=\bigoplus _{i}Z(\mathbf {M} )P_{i}}$

where each Z(M)P_i is a commutative simple algebra. Every complex simple algebras is isomorphic to the full matrix algebra C^{k × k} for some k. But Z(M)P_i is commutative, therefore one-dimensional.

The projections P_i "diagonalizes" M in a natural way. For M ∈ M, M can be uniquely decomposed into M = Σ MP_i. Therefore,

${\displaystyle {\mathbf {M} }=\bigoplus _{i}{\mathbf {M} }P_{i}.}$

One can see that Z(MP_i) = Z(M)P_i. So Z(MP_i) is one-dimensional and each MP_i is a factor. This proves the claim.

For general von Neumann algebras, the direct sum is replaced by the direct integral. The above is a special case of the central decomposition of von Neumann algebras.