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 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.