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  "assumptions": "Let A be a nonzero unital complex C*-algebra, D a closed unital star subalgebra of A, and χ a character of D.",
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      "label": "MathlibAnnex.Analysis.CStarAlgebra.IsPureState",
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  "conclusion": "There is a continuous complex-linear functional φ on A in stateSpace A that is pure and restricts to χ: for every d in D, φ(d)=χ(d).",
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  "display_title": "Pure-state extension of a character",
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  "name": "MathlibAnnex.Analysis.CStarAlgebra.exists_pureState_extension",
  "natural_language_statement": "Let A be a nonzero unital complex C*-algebra, D a closed unital star subalgebra of A, and χ a character of D. There is a continuous complex-linear functional φ on A in stateSpace A that is pure and restricts to χ: for every d in D, φ(d)=χ(d).",
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      "text": "Exact Lean statement: ∀ {A : Type u} [inst : CStarAlgebra A] [inst_1 : PartialOrder A] [inst_2 : StarOrderedRing A] [Nontrivial A] (D : StarSubalgebra ℂ A) [inst_4 : IsClosed ↑D] (chi : ↑(WeakDual.characterSpace ℂ ↥D)), ∃ phi ∈ MathlibAnnex.Analysis.CStarAlgebra.stateSpace A, MathlibAnnex.Analysis.CStarAlgebra.IsPureState A phi ∧ ∀ (d : ↥D), phi ↑d = chi d"
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      "text": "∃ φ : A →L[ℂ] ℂ, φ ∈ stateSpace A ∧ IsPureState A φ ∧ ∀ d : D, φ d = χ d."
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      "text": "The source obtains an extreme point of the compact weak state-extension face. It maps that weak functional to the strong dual, preserves its restriction to D, and transports extremality to show purity. Closedness of D is used by the extension-face construction."
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  "proof_summary": "The source obtains an extreme point of the compact weak state-extension face. It maps that weak functional to the strong dual, preserves its restriction to D, and transports extremality to show purity. Closedness of D is used by the extension-face construction.",
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    "text": "theorem exists_pureState_extension (D : StarSubalgebra ℂ A) [IsClosed (D : Set A)]\n    (chi : WeakDual.characterSpace ℂ D) :\n    ∃ phi : A →L[ℂ] ℂ,\n      phi ∈ stateSpace A ∧ IsPureState A phi ∧ ∀ d : D, phi d = chi d"
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