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  "assumptions": "Let D be a commutative unital complex C*-algebra, J an ideal whose carrier is norm-closed, and d∈D with d∉J.",
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  "display_title": "A character separates a closed ideal",
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  "name": "MathlibAnnex.Analysis.CStarAlgebra.exists_character_annihilating_closedIdeal_of_not_mem",
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      "text": "Exact Lean statement: ∀ {D : Type u} [inst : CommCStarAlgebra D] (J : Ideal D), IsClosed ↑J → ∀ {d : D}, d ∉ J → ∃ chi, (∀ x ∈ J, chi x = 0) ∧ chi d ≠ 0"
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      "text": "∃ χ, (∀ x : D, x∈J → χ x=0) ∧ χ d≠0."
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      "text": "The source transports J through the Gelfand star transform to an ideal of continuous functions, uses isometry to preserve closedness and d∉J, then separates the transformed point by a character and pulls the annihilation/nonvanishing equations back."
    }
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    "text": "theorem exists_character_annihilating_closedIdeal_of_not_mem\n    {D : Type u} [CommCStarAlgebra D]\n    (J : Ideal D) (hJclosed : IsClosed (J : Set D))\n    {d : D} (hd : d ∉ J) :\n    ∃ chi : WeakDual.characterSpace ℂ D,\n      (∀ x : D, x ∈ J → chi x = 0) ∧ chi d ≠ 0"
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