GNS Construction

graph TD D1["Def: State on C*\nω:A→C linear pos"] D2["Def: GNS rep\nH_ω from ω, π_ω"] D3["Def: Cyclic vector\nξ, π(A)ξ dense"] D4["Def: Faithful state\nω(x*x)=0 ⇒ x=0"] T1["Thm: GNS unique\nω → (H_ω,π_ω,ξ) unique"] T2["Thm: Cyclic\nπ_ω cyclic for ξ"] T3["Thm: ω(x)=⟨ξ,π(x)ξ⟩\nrecovery formula"] T4["Thm: Irreducible iff\nω pure"] T5["Thm: Universal rep\n⊕ all GNS reps"] D1 --> D2 D2 --> D3 D1 --> D4 D2 --> T1 D3 --> T2 T1 --> T3 D4 --> T4 T1 --> T5 classDef definition fill:#b197fc,color:#fff classDef theorem fill:#51cf66,color:#fff class D1,D2,D3,D4 definition class T1,T2,T3,T4,T5 theorem