Finite dimensional von Neumann algebra

In mathematics, von Neumann algebras are self-adjoint operator algebras that are closed under a chosen operator topology. In Mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the In Functional analysis, an operator algebra is an algebra of continuous Linear operators on a Topological vector space with the multiplication In Mathematics, the requirements of Functional analysis mean there are several standard topologies which are given to the algebra B ( H) of 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. Joseph Henry Maclagan Wedderburn ( 2 February 1882 Forfar Angus, Scotland – 9 October 1948, Princeton New Jersey In Ring theory, a semisimple algebra is an Associative algebra which has trivial Jacobson radical (that is only the zero element of the algebra is in the

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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. In Ring theory, a semisimple algebra is an Associative algebra which has trivial Jacobson radical (that is only the zero element of the algebra is in the 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.

Let The center of a von Neumann algebra M will be denoted by Z(M). The term center or centre is used in various contexts in Abstract algebra to denote the set of all those elements that commute with all other elements Since M is self-adjoint, Z(M) is itself a (commutative) von Neumann algebra. A von Neumann algebra N is called a factor is 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. In Mathematics and Functional analysis a direct integral is a generalization of the concept of Direct sum. The above is a special case of the central decomposition of von Neumann algebras. In Mathematics and Functional analysis a direct integral is a generalization of the concept of Direct sum.

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