almost all x ∈ X, Ax is a von Neumann algebra that is a factor. 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 πx, πy 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 πx belongs to the class of x. This decomposition is essentially unique. This result is fundamental in the theory of group representations. References J. Dixmier, Von Neumann algebras, J. Dixmier, C* algebras G. W. Mackey, The Theory of Unitary Group Representations, The University of Chicago Press, 1976. J. von Neumann, On Rings of Operators. Reduction Theory The Annals of Mathematics 2nd Ser., Vol. 50, No. 2 (Apr., 1949), pp. 401–485. Masamichi Takesaki ''Theory of Operator Algebras I,II,III", encyclopedia of mathematical sciences, Springer-Verlag, 2001–2003 (the first volume was published 1979 in 1. Edition) Functional analysis Measure theory Von Neumann algebras