I have never been happy with the fact a single quantum state could be encoded by an infinite number of vectors [itex]|\phi\rangle[/itex]. Choosing a unit vector limits this overabundance but you have still an infinity of (physically equivalent) possibilities left. I later realized that the projector [itex]|\phi\rangle\langle\phi|[/itex] was unique and had the other advantage of being of the same kind as mixed states. If observable are operators and states can be represented (and in a better way) by operators, then fine, let us speak about operators only! Furthermore if we do not care about what the operators are applied to, let us just forget the whole idea of operators and concentrate on their mutual relations. That's where I discovered C* algebras. In the end I'm wondering if there are good reasons to keep the idea of Hilbert space or if only its historical primacy keeps its widely used. The only limitation of c* algebra I'm aware of (but I'm really more aware of the mathematical structure than its use in qm) is that they can only represent bounded operators whereas an Hilbert space can be extended to a rigged Hilbert space to deal rigorously with that problem. Are there some known similar structures that extend c*-algebras and address the same issue ? Are they other limitations and situations where Hilbert spaces still remain better tools ?