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  "assumptions": "Let A be a complex C*-algebra, not assumed unital, H a complex Hilbert space, and ρ a representation of A on H.",
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  "conclusion": "compactPreimageIdeal ρ is the two-sided algebraic ideal of elements whose represented bounded operator is compact. Norm-closedness is a separate theorem and is not a field of this returned TwoSidedIdeal.",
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  "name": "MathlibAnnex.CStarAlgebra.compactPreimageIdeal",
  "natural_language_statement": "Let A be a complex C*-algebra, not assumed unital, H a complex Hilbert space, and ρ a representation of A on H. compactPreimageIdeal ρ is the two-sided algebraic ideal of elements whose represented bounded operator is compact. Norm-closedness is a separate theorem and is not a field of this returned TwoSidedIdeal.",
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      "text": "Exact Lean statement: {A : Type u} → {H : Type v} → [inst : NormedAddCommGroup H] → [inst_1 : InnerProductSpace ℂ H] → [inst_2 : CompleteSpace H] → [inst_3 : NonUnitalCStarAlgebra A] → (A →⋆ₙₐ[ℂ] H →L[ℂ] H) → TwoSidedIdeal A"
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      "text": "(compactPreimageIdeal ρ : Set A) = {a : A | IsCompactOperator (ρ a)}."
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      "text": "The source constructs TwoSidedIdeal.mk' with carrier given by inverse image of compactOperator, checking zero, addition, negation and left/right multiplication using compact-operator composition. A later isClosed_compactPreimageIdeal establishes topological closedness."
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    "text": "def compactPreimageIdeal [NonUnitalCStarAlgebra A]\n    (rho : A →⋆ₙₐ[ℂ] (H →L[ℂ] H)) : TwoSidedIdeal A"
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