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 formula_1, then formula_2. 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.
Properties.
Let formula_3 denote a finite von Neumann algebra with center formula_4. One of the fundamental characterizing properties of finite von Neumann algebras is the existence of a center-valued trace. This is a normal positive bounded map formula_5 with the properties:
Examples.
Finite-dimensional von Neumann algebras.
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) {"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 C"k" × "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.