Hilbert spaces and quantum mechanics

Question

1. how is Hilbert spaces applied in quantum mechanics?
2. the differences between the application of C* -algebra and Hilbert spaces on quantum mechanics.

Answer 1

In quantum mechanics the state of a physical system is represented by a vector in a Hilbert space. A physical observable is a self-adjoint operator on a Hilbert space.

So elements in a certain Hilbert space correspond to states of a physical system whereas physical observables can be described by operators on that Hilbert space.

A $C^*$-algebra is a certain subalgebra of the bounded operators on some Hilbert space $(B(\mathcal{H}))$. Hence an element in a $C^*$-algebra corresponds to a physical observable.