Editing Direct integral (section)

== Measurable families of representations ==

If ''A'' is a separable [[C*-algebra]], the above results can be applied to measurable families of non-degenerate *-representations of ''A''. In the case that ''A'' has a unit, non-degeneracy is equivalent to unit-preserving. By the general correspondence that exists between [[strongly continuous]] [[unitary representation]]s of a [[locally compact group]] ''G'' and non-degenerate *-representations of the groups C*-algebra C*(''G''), the theory for C*-algebras immediately provides a decomposition theory for representations of [[separable space|separable]] locally compact groups.

'''Theorem'''. Let ''A'' be a separable C*-algebra and π a non-degenerate involutive representation of ''A'' on a separable Hilbert space ''H''. Let W*(π) be the von Neumann algebra generated by the operators π(''a'') for ''a'' ∈ ''A''. Then corresponding to any central decomposition of W*(π) over a standard measure space (''X'', μ) (which, as stated, is unique in a measure theoretic sense), there is a measurable family of factor representations

:<math> \{\pi_x\}_{x \in X} </math>

of ''A'' such that

:<math> \pi(a) = \int_X^\oplus \pi_x(a) d \mu(x), \quad \forall a \in A. </math>

Moreover, there is a subset ''N'' of ''X'' with μ measure zero, such that π<sub>''x''</sub>, π<sub>''y''</sub> are disjoint whenever ''x'', ''y'' ∈ ''X'' − ''N'', where representations are said to be ''disjoint'' if and only if there are no [[intertwining operator]]s between them.

One can show that the direct integral can be indexed on the so-called ''quasi-spectrum'' ''Q'' of ''A'', consisting of quasi-equivalence classes of factor representations of ''A''. 
Thus, there is a standard measure μ on ''Q'' and a measurable family of factor representations indexed on ''Q'' such that π<sub>''x''</sub> belongs to the class of ''x''. This decomposition is essentially unique. This result is fundamental in the theory of [[group representation]]s.