I think Strocchi's book An introduction to the Mathematical structure of Quantum Mechanics may do the job. In particular the first two chapters.

I think Strocchi's book An introduction to the Mathematical structure of Quantum Mechanics may do the job. In particular section 1.3 explains what you want (states, observables, expectations, $C^*$-algebras and representations). The discussion is carried out in more detail in the first two chapters.