Talk:Spectrum of a C*-algebra

Little question

Can a representation always be written as an integral of irreducible ones? I've heard about "integral representation of states" and barycentric decomposition but didn't have the time to check it out.

What I've seen is the decomposition of a (concrete) von Neumann algebra acting on a separable Hilbert space as a direct integral of factors. But factors of type II and III (which appear in representations of so called non-GCR or antiliminal $C^*$-algebras) do not have irreducible subrepresentations, which made me think it was impossible to write an arbitrary representation of a $C^*$-algebra from irreducible representations.