I have been thinking about what can we say about decomposing any representation as a direct sum of irreducible representations. I know that every cyclic representation corresponds to a representation coming from a state, which is the essence of the GNS construction. Moreover such a representation is irreducible $\textbf{iff}$ the state is a pure state.
I also know that every representation can be decomposed into a direct sum of cyclic representations. Now my question is whether we can decompose any representation of a C* algebra $A$ into direct sum of irreducible representations. It seems to me that not every representation can be represenation can be decomposed in this way. Is there a description/classification of those representations which can be decomposed in this way.
I don't know much representation theory but I think such results are true for representations of some other class of objects.