Direct Integral - Central Decomposition

Central Decomposition

Suppose A is a von Neumann algebra. let Z(A) be the center of A, that is the set of operators in A that commute with all operators A, that is

Z(A) is an Abelian von Neumann algebra.

Example. The center of L(H) is 1-dimensional. In general, if A is a von Neumann algebra, if the center is 1 dimensional we say A is a factor.

Now suppose A is a von Neumann algebra whose center contains a sequence of minimal pairwise orthogonal non-zero projections {E_i}_{i ∈ N} such that

Then A E_i is a von Neumann algebra on the range H_i of E_i. It is easy to see A E_i is a factor. Thus in this special case

represents A as a direct sum of factors. This is a special case of the central decomposition theorem of von Neumann.

In general, we can apply the structure theorem of Abelian von Neumann algebras that represents Z(A) as an algebra of scalar diagonal operators. In any such representation, all the operators in A are decomposable operators. In fact, we can use this to prove the basic result of von Neumann that any von Neumann algebra admits a decomposition into factors.

Theorem. Suppose

is a direct integral decomposition of H and A is a von Neumann algebra on H so that Z(A) is represented by the algebra of scalar diagonal operators L∞_μ(X) where X is a standard Borel space. Then

where for almost all x ∈ X, A_x is a von Neumann algebra that is a factor.