Direct Integral - Measurable Families of Representations

Measurable Families of Representations

If A is a separable C*-algebra, we can consider measurable families of non-degenerate *-representations of A; recall that in case A has a unit, non-degeneracy is equivalent to unit-preserving. By the general correspondence that exists between strongly continuous unitary representations 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 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

of A such that

Moreover, there is a subset N of X with μ measure zero, such that π_x, π_y are disjoint whenever x, y ∈ X − N, where representations are said to be disjoint if and only if there are no intertwining operators 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_x belongs to the class of x. This decomposition is essentially unique. This result is fundamental in the theory of group representations.