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"(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.