Von Neumann Algebras

graph TD D1["Def: Von Neumann algebra\nM⊂B(H), M=M''"] D2["Def: Commutant\nM' = {x: xm=mx ∀m}"] D3["Def: Factor\nM∩M' = C1"] D4["Def: Type I, II, III\nclassification by proj lattice"] T1["Thm: Bicommutant\nM''=strong cl(M)"] T2["Thm: Double comm.\nM = M''"] T3["Thm: Type decompos.\nany vN → factors"] T4["Thm: Tomita–Takesaki\nmodular theory"] T5["Thm: Murray–von Neumann\nfactors type I,II,III"] D2 --> D1 D1 --> T1 T1 --> T2 D1 --> D3 D3 --> D4 D4 --> T5 D1 --> T3 T3 --> T4 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