In mathematics, a finite von Neumann algebra is a von Neumann algebra in which every isometry is a unitary. In other words, for an operator V in a finite von Neumann algebra if [math]\displaystyle{ V^*V = I }[/math], then [math]\displaystyle{ VV^* = I }[/math]. In terms of the comparison theory of projections, the identity operator is not (Murray-von Neumann) equivalent to any proper subprojection in the von Neumann algebra.
Let [math]\displaystyle{ \mathcal{M} }[/math] denote a finite von Neumann algebra with center [math]\displaystyle{ \mathcal{Z} }[/math]. One of the fundamental characterizing properties of finite von Neumann algebras is the existence of a center-valued trace. A von Neumann algebra [math]\displaystyle{ \mathcal{M} }[/math] is finite if and only if there exists a normal positive bounded map [math]\displaystyle{ \tau : \mathcal{M} \to \mathcal{Z} }[/math] with the properties:
The finite-dimensional von Neumann algebras can be characterized using Wedderburn's theory of semisimple algebras. Let Cn × n be the n × n matrices with complex entries. A von Neumann algebra M is a self adjoint subalgebra in Cn × n such that M contains the identity operator I in Cn × 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) {Pi} such that PiPj = 0 for i ≠ j, Σ Pi = I, and
where each Z(M)Pi is a commutative simple algebra. Every complex simple algebras is isomorphic to the full matrix algebra Ck × k for some k. But Z(M)Pi is commutative, therefore one-dimensional.
The projections Pi "diagonalizes" M in a natural way. For M ∈ M, M can be uniquely decomposed into M = Σ MPi. Therefore,
One can see that Z(MPi) = Z(M)Pi. So Z(MPi) is one-dimensional and each MPi 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.
Original source: https://en.wikipedia.org/wiki/Finite von Neumann algebra.
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