Direct Integrals of Unitarily Equivalent Representations of Nonseparable C*-Algebras

Let π0 be a fixed irreducible representation of a C*algebra A and π = ∫⊕Z π0dμ(ζ) a direct integral of representations of A, each of which is a copy of π0. If A is separable, π is a multiple of π0. For nonseparable C*-algebras Baggett and Ramsay found a counterexample. We generalize this example and show the following result: π is a multiple of π0, if there is a cyclic vector ξ = ∫⊕Z ξ(ζ) dμ(ζ) for π and a negligible set N in Z such that the Hilbert space generated by {ξ(ζ) : ζ ∈ Z\N} is separable. Under additional assumptions this condition is also necessary for π being a multiple of π0.