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  "assumptions": "Let A be a unital complex C*-algebra and let φ be a pure state of A.",
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  "conclusion": "The canonical GNS star-algebra homomorphism built from the positive map associated to φ is irreducible.",
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  "name": "MathlibAnnex.Analysis.CStarAlgebra.isIrreducible_pureState_gnsStarAlgHom",
  "natural_language_statement": "Let A be a unital complex C*-algebra and let φ be a pure state of A. The canonical GNS star-algebra homomorphism built from the positive map associated to φ is irreducible.",
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      "text": "Exact Lean statement: ∀ {A : Type u} [inst : CStarAlgebra A] [inst_1 : PartialOrder A] [inst_2 : StarOrderedRing A] (phi : A →L[ℂ] ℂ) (hphi : phi ∈ MathlibAnnex.Analysis.CStarAlgebra.stateSpace A), MathlibAnnex.Analysis.CStarAlgebra.IsPureState A phi → (MathlibAnnex.Analysis.CStarAlgebra.positiveLinearMapOfMemStateSpace phi hphi).gnsStarAlgHom.IsIrreducible"
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      "text": "IsIrreducible ((positiveLinearMapOfMemStateSpace φ hφ).gnsStarAlgHom)."
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      "text": "The source invokes the cyclic-vector irreducibility criterion: the GNS cyclic vector has norm one and dense orbit, its vector functional equals φ by the GNS inner-product identity, and purity supplies the criterion's extremality premise."
    }
  ],
  "proof_summary": "The source invokes the cyclic-vector irreducibility criterion: the GNS cyclic vector has norm one and dense orbit, its vector functional equals φ by the GNS inner-product identity, and purity supplies the criterion's extremality premise.",
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    "text": "theorem isIrreducible_pureState_gnsStarAlgHom\n    (phi : A →L[ℂ] ℂ) (hphi : phi ∈ stateSpace A) (hpure : IsPureState A phi) :\n    StarAlgHom.IsIrreducible\n      (positiveLinearMapOfMemStateSpace phi hphi).gnsStarAlgHom"
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