C*-Algebras

graph TD D1["Def: C*-algebra\nBanach *-alg, ‖x*x‖=‖x‖²"] D2["Def: Commutative C*\nA ≅ C₀(X) locally compact"] D3["Def: Positive element\nx ≥ 0 iff x = a*a"] D4["Def: State\nω linear, ω(1)=1, ω(x*x)≥0"] T1["Thm: Gelfand–Naimark\nA↪B(H) isometry"] T2["Thm: GNS\nstate → cyclic rep"] T3["Thm: Spectrum\nσ(x) compact in C"] T4["Thm: Continuous f.calc\nf(x) well-defined"] T5["Thm: Spectral radius\nr(x)=‖x‖ for normal"] D1 --> D2 D1 --> D3 D3 --> D4 D4 --> T2 D1 --> T1 D1 --> T3 T3 --> T4 T3 --> 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