Finite-dimensional von Neumann algebra

In mathematics, von Neumann algebras are self-adjoint operator algebras that are closed under a chosen operator topology. When the underlying Hilbert space is finite-dimensional, the von Neumann algebra is said to be a finite-dimensional von Neumann algebra. The finite-dimensional case differs from the general von Neumann algebras in that topology plays no role and they can be characterized using Wedderburn's theory of semisimple algebras.

Details

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

$Z(\mathbf M) = \oplus _i Z(\mathbf M) P_i$

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,

${\mathbf M} = \oplus_i {\mathbf M} P_i .$

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