Editing Direct integral (section)

== Central decomposition ==

Suppose ''A'' is a von Neumann algebra. Let '''Z'''(''A'') be the [[center (algebra)|center]] of ''A''. The center is the set of operators in ''A'' that commute with all operators ''A'':

:<math> \mathbf{Z}(A) = A \cap A' </math>

Then '''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'''.

When ''A'' is a von Neumann algebra whose center contains a sequence of minimal pairwise orthogonal non-zero projections {''E''<sub>''i''</sub>}<sub>''i'' ∈ '''N'''</sub> such that

:<math> 1 = \sum_{i \in \mathbb{N}} E_i </math>

then ''A'' ''E''<sub>''i''</sub> is a von Neumann algebra on the range ''H''<sub>''i''</sub> of ''E''<sub>''i''</sub>. It is easy to see ''A'' ''E''<sub>''i''</sub> is a factor. Thus, in this special case

:<math> A = \bigoplus_{i \in \mathbb{N}} A E_i </math>

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

In general, the structure theorem of Abelian von Neumann algebras represents Z('''A''') as an algebra of scalar diagonal operators. In any such representation, all the operators in '''A''' are decomposable operators. This can be used to prove the basic result of von Neumann: any von Neumann algebra admits a decomposition into factors.

'''Theorem'''. Suppose

:<math> H = \int_X^\oplus H_x d \mu(x) </math>

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''<sup>∞</sup><sub>μ</sub>(''X'') where ''X'' is a standard Borel space. Then

:<math> \mathbf{A} = \int^\oplus_X A_x d \mu(x) </math>

where for almost all ''x'' ∈ ''X'', ''A''<sub>''x''</sub> is a von Neumann algebra that is a ''factor''.