Importance of pure states in $C^{*}$ algebras

Question

In a tensor product of $C^{*}$-algebras I have seen some proofs are used by the argument of norms by pure states. Why pure states are essential to study related to vN algebras and $C^{*}$-algebras?

Answer 1

The question is fairly broad, but here are a couple things:

• the pure states are the extreme points of the state space. So any property that holds for pure states and survives convex combinations and pointiwse limits, holds for all states.

• pure states are "minimal", in the sense that if $\psi\leq\phi$ with both states and $\phi$ pure, then $\psi=c\,\phi$ for some $c>0$.

• the GNS representation of a pure state is irreducible.

Without thinking of a specific case, this is as far as I can go.