Direct integrals of unitarily equivalent representations of nonseparable \(C^*\)-algebras (Q1803339)

Let \(\pi_ 0\) be a representation of a separable \(C^*\)-algebra. It is known that if \(\pi\) is a direct integral of representations unitarily equivalent to \(\pi_ 0\), then \(\pi\) is a direct sum of copies of \(\pi_ 0\). Due to an example of Baggett and Ramsay, this result does not extend to the nonseparable case. The aim of the paper is to discuss the nonseparable case, keeping \(\pi_ 0\) irreducible. The author shows that if there is a cyclic vector \(\int_ Z^ \oplus \zeta(z) d\mu(z)\) for \(\pi\) and a negligible set \(N\subset Z\) such that \(\{\zeta(z)\): \(\in Z\setminus N\}\) generates a separable Hilbert space, then \(\pi\) is a multiple of \(\pi_ 0\).