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 $V^*V = I$, then $VV^* = I$. 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 $\mathcal$ denote a finite von Neumann algebra with center $\mathcal$. 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 $\tau : \mathcal \to \mathcal$ with the properties:
* $\tau(AB) = \tau(BA), A, B \in \mathcal$,
* if $A \ge 0$ and $\tau(A) = 0$ then $A = 0$,
* $\tau(C) = C$ for $C \in \mathcal$,
* $\tau(CA) = C\tau(A)$ for $A \in \mathcal$ and $C \in \mathcal$.

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) such that ''P_iP_j'' = 0 for ''i'' ≠ ''j'', Σ ''P_i'' = ''I'', and

:$Z(\mathbf M) = \bigoplus _i Z(\mathbf M) P_i$

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,

:$= \bigoplus_i P_i .$

One can see that ''Z''(M''P_i'') = ''Z''(M'')P_i''. So ''Z''(M''P_i'') is one-dimensional and each M''P_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.

Abelian von Neumann algebras

Type $II_1$ factors

References

*

*

{{DEFAULTSORT:Finite von Neumann Algebra
Linear algebra
*