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  "assumptions": "Let A be a nonzero complex C*-algebra, not assumed unital; let H be a separable 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; and let D be a closed unital star subalgebra of Unitization ℂ A.",
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      "label": "MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation.exists_unit_eigenvector_of_character_ne_infinity",
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      "text": "Exact formal dependency; inspect the linked Card and exact source."
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  "conclusion": "The entire character space of D is countable, including the distinguished scalar character.",
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  "definition_equation": "Countable (WeakDual.characterSpace ℂ D).",
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  "exact_elaborated_lean_type": "∀ {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] [TopologicalSpace.SeparableSpace H] (pi : MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation A H), pi.IsSingletonIrreducibleModel → ∀ (D : StarSubalgebra ℂ (Unitization ℂ A)) [inst_8 : IsClosed ↑D], Countable ↑(WeakDual.characterSpace ℂ ↥D)",
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  "name": "MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation.countable_characterSpace_of_nonUnital_singleton",
  "natural_language_statement": "Let A be a nonzero complex C*-algebra, not assumed unital; let H be a separable 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; and let D be a closed unital star subalgebra of Unitization ℂ A. The entire character space of D is countable, including the distinguished scalar character.",
  "one_sentence_role": "Shows that the full character space of a closed unitization subalgebra is countable when π is a separably acting representative of the unique irreducible-representation class.",
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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] [TopologicalSpace.SeparableSpace H] (pi : MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation A H), pi.IsSingletonIrreducibleModel → ∀ (D : StarSubalgebra ℂ (Unitization ℂ A)) [inst_8 : IsClosed ↑D], Countable ↑(WeakDual.characterSpace ℂ ↥D)"
    },
    {
      "step_id": "source_equation",
      "text": "Countable (WeakDual.characterSpace ℂ D)."
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    {
      "step_id": "source_route",
      "text": "For each character other than infinityCharacterOn D, the prior source theorem supplies a joint unit eigenvector. Distinct characters give orthogonal vectors, so separability of H makes that complement countable. The source then maps the disjoint sum of the complement and one Unit onto the entire character space."
    }
  ],
  "proof_summary": "For each character other than infinityCharacterOn D, the prior source theorem supplies a joint unit eigenvector. Distinct characters give orthogonal vectors, so separability of H makes that complement countable. The source then maps the disjoint sum of the complement and one Unit onto the entire character space.",
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      "source_path": "MathlibAnnex/Analysis/CStarAlgebra/NonUnital/CharacterCountable.lean",
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    "start_byte": 3844,
    "text": "theorem countable_characterSpace_of_nonUnital_singleton [Nontrivial A]\n    [TopologicalSpace.SeparableSpace H]\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    Countable (WeakDual.characterSpace ℂ D)"
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