Spectral Theory for C*-Algebras

graph TD D1["Def: Spectrum σ(x)\nλ: x−λ1 not inv"] D2["Def: Spectral radius\nr(x)=max|σ(x)|"] D3["Def: Normal element\nx*x=xx*"] D4["Def: C*-funct. calc\nf(x) for f∈C(σ(x))"] T1["Thm: σ(x) nonempty\ncompact in C"] T2["Thm: r(x)=lim ‖xⁿ‖^{1/n}"] T3["Thm: Spectral mapping\nσ(f(x))=f(σ(x))"] T4["Thm: Continuous calc\nf↦f(x) iso"] T5["Thm: Borel calc\nextends to bounded Borel"] D1 --> D2 D1 --> D3 D3 --> D4 D1 --> T1 D2 --> T2 D4 --> T3 T3 --> T4 T4 --> 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