I think Strocchi's book [An introduction to the Mathematical structure of Quantum Mechanics][1]  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. 



  [1]: https://www.worldscientific.com/worldscibooks/10.1142/7038