Positivity, Representations, Tensor Products, and Ideals in C*-Algebras

In classical mechanics an observable is a continuous function on a space X and an element of the commutative C*-algebra \(C(X)\) . In quantum mechanics an observable is a bounded operator on a Hilbert space, or an element of a C*-subalgebra \(A\subset \mathbb {B} (H)\) , such as \(M_n(\mathbb {C})\) , if the system is very simple, like that of an atom which can be in n quantum states, corresponding to energy levels of the Hamiltonian. A “microscopic” state of the classical system is a point of the space and corresponds to a C*-algebra character \(C(X) \to \mathbb {C}\) . In quantum mechanics one is forced to work with general states in the C*-algebraic sense: positive linear functionals \(\varphi \colon A \to \mathbb {C}\) , i.e., linear functionals taking positive values at self-adjoint elements of A with positive spectrum. For example, the the energy states of the Hamiltonian are the vector states corresponding to the eigenfunctions of the Hamiltonian H.