A mixed state Ω: A → ℂ on a C*-algebra A is a state that can be written as a convex linear sum of states Ω1, Ω2 such that
Ω = λΩ1 + (1 - λ)Ω2, O < λ < 1, and a state Ω is pure if it is not mixed.
Given the GNS representation πΩ: A → B(H) on a Hilbert space H based at a reference vector ΨΩ, can you show that the representation πΩ is irreducible if and only if Ω is pure?