Finite von Neumann algebra
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, then. In terms of the von Neumann algebra#Comparison [theory of projections|comparison theory of projections], the identity operator is not equivalent to any proper subprojection in the von Neumann algebra.
Properties
Let denote a finite von Neumann algebra with center. One of the fundamental characterizing properties of finite von Neumann algebras is the existence of a center-valued trace. A von Neumann algebra is finite if and only if there exists a normal positive bounded map with the properties:- ,
- if and then,
- for,
- for and.
Examples
Finite-dimensional von Neumann algebras
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 C'n × 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 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. Since M is self-adjoint, Z is itself a von Neumann algebra. A von Neumann algebra N is called a factor if Z is one-dimensional, that is, Z 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.
Proof: By Wedderburn's theory of semisimple algebras, Z contains a finite orthogonal set of idempotents such that PiPj = 0 for i ≠ j, Σ Pi = I, and
where each ZPi is a commutative simple algebra. Every complex simple algebras is isomorphic to
the full matrix algebra Ck'' × k for some k. But ZPi 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 = ZPi. So Z'' 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.