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  "assumptions": "Let A be a nonzero complex C*-algebra, not assumed unital; let H be a complex Hilbert space; let π be a representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A; let D be a closed unital star subalgebra of Unitization ℂ A; and let χ be a character of D distinct from infinityCharacterOn D. No separability assumption on H is required.",
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  "conclusion": "There is a unit vector η in H such that π.unitization(d)η=χ(d)η for every d in D.",
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  "display_title": "Eigenvector for a character distinct from the scalar character",
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  "name": "MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation.exists_unit_eigenvector_of_character_ne_infinity",
  "natural_language_statement": "Let A be a nonzero complex C*-algebra, not assumed unital; let H be a complex Hilbert space; let π be a representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A; let D be a closed unital star subalgebra of Unitization ℂ A; and let χ be a character of D distinct from infinityCharacterOn D. No separability assumption on H is required. There is a unit vector η in H such that π.unitization(d)η=χ(d)η for every d in D.",
  "one_sentence_role": "Produces a unit joint eigenvector whose eigenvalue character is the prescribed character distinct from the scalar character.",
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      "step_id": "formal_type",
      "text": "Exact Lean statement: ∀ {A : Type u} [inst : NonUnitalCStarAlgebra A] [inst_1 : PartialOrder A] [StarOrderedRing A] {H : Type v} [inst_3 : NormedAddCommGroup H] [inst_4 : InnerProductSpace ℂ H] [inst_5 : CompleteSpace H] [Nontrivial A] (pi : MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation A H), pi.IsSingletonIrreducibleModel → ∀ (D : StarSubalgebra ℂ (Unitization ℂ A)) [inst_7 : IsClosed ↑D] (chi : ↑(WeakDual.characterSpace ℂ ↥D)), chi ≠ MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation.infinityCharacterOn D → ∃ eta, ‖eta‖ = 1 ∧ ∀ (d : ↥D), (pi.unitization ↑d) eta = chi d • eta"
    },
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      "text": "∃ η : H, ‖η‖=1 ∧ ∀ d : D, π.unitization d η = χ d • η."
    },
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      "text": "The source extends χ to a pure state on the unitization, forms its GNS representation, and shows its restriction to A is nonzero because χ is distinct from the scalar character. The unique-class hypothesis gives unitary equivalence to π and transfers the normalized GNS cyclic vector to η, preserving the joint eigenvector equation."
    }
  ],
  "proof_summary": "The source extends χ to a pure state on the unitization, forms its GNS representation, and shows its restriction to A is nonzero because χ is distinct from the scalar character. The unique-class hypothesis gives unitary equivalence to π and transfers the normalized GNS cyclic vector to η, preserving the joint eigenvector equation.",
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    "text": "theorem exists_unit_eigenvector_of_character_ne_infinity [Nontrivial A]\n    (pi : NonUnitalCStarRepresentation A H)\n    (hsingle : IsSingletonIrreducibleModel.{u, v, u} pi)\n    (D : StarSubalgebra ℂ (Unitization ℂ A))\n    [IsClosed (D : Set (Unitization ℂ A))]\n    (chi : WeakDual.characterSpace ℂ D)\n    (hchi : chi ≠ infinityCharacterOn (A := A) D) :\n    ∃ eta : H, ‖eta‖ = 1 ∧\n      ∀ d : D, pi.unitization (d : Unitization ℂ A) eta = chi d • eta"
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