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  "assumptions": "Let A be a complex C*-algebra, not assumed unital. Let D be a closed commutative unital star subalgebra of Unitization ℂ A with countable character space, and let 0 ≠ d ∈ D have zero scalar coordinate.",
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      "label": "MathlibAnnex.Topology.exists_isOpen_singleton",
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  "conclusion": "There exists a character χ distinct from the restricted infinity character for which {χ} is open in the character space. This theorem itself does not assume that any representation represents a unique irreducible-representation class.",
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  "definition_equation": "∃ χ, χ ≠ infinityCharacterOn D ∧ IsOpen {χ}.",
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  "display_title": "Isolated character away from infinity",
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  "name": "MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation.exists_isolated_character_ne_infinity",
  "natural_language_statement": "Let A be a complex C*-algebra, not assumed unital. Let D be a closed commutative unital star subalgebra of Unitization ℂ A with countable character space, and let 0 ≠ d ∈ D have zero scalar coordinate. There exists a character χ distinct from the restricted infinity character for which {χ} is open in the character space. This theorem itself does not assume that any representation represents a unique irreducible-representation class.",
  "one_sentence_role": "Finds an isolated character distinct from the scalar character in a countable character space.",
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      "text": "Exact Lean statement: ∀ {A : Type u} [inst : NonUnitalCStarAlgebra A] [inst_1 : PartialOrder A] [StarOrderedRing A] (D : StarSubalgebra ℂ (Unitization ℂ A)) [inst_3 : IsClosed ↑D] [inst_4 : IsMulCommutative ↥D] [Countable ↑(WeakDual.characterSpace ℂ ↥D)] (d : ↥D), d ≠ 0 → (↑d).toProd.1 = 0 → ∃ chi, chi ≠ MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation.infinityCharacterOn D ∧ IsOpen {chi}"
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      "step_id": "source_equation",
      "text": "∃ χ, χ ≠ infinityCharacterOn D ∧ IsOpen {χ}."
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    {
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      "text": "Gelfand-transform injectivity finds a character not annihilating d; the zero scalar coordinate excludes the infinity character. The complement of that closed singleton is a nonempty open Baire subspace. The countable T1 Baire lemma yields an isolated point there, and openness transfers back to the full character space."
    }
  ],
  "proof_summary": "Gelfand-transform injectivity finds a character not annihilating d; the zero scalar coordinate excludes the infinity character. The complement of that closed singleton is a nonempty open Baire subspace. The countable T1 Baire lemma yields an isolated point there, and openness transfers back to the full character space.",
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    "text": "theorem exists_isolated_character_ne_infinity\n    (D : StarSubalgebra ℂ (Unitization ℂ A))\n    [IsClosed (D : Set (Unitization ℂ A))]\n    [IsMulCommutative D]\n    [Countable (WeakDual.characterSpace ℂ D)]\n    (d : D) (hd : d ≠ 0)\n    (hdfst : (d : Unitization ℂ A).fst = 0) :\n    ∃ chi : WeakDual.characterSpace ℂ D,\n      chi ≠ infinityCharacterOn (A := A) D ∧\n        IsOpen ({chi} : Set (WeakDual.characterSpace ℂ D))"
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