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  "assumptions": "Let A be a nonzero complex C*-algebra, not assumed unital, H a separable complex Hilbert space, and π a representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. No separability of A, prior faithfulness, or simplicity is assumed.",
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      "label": "MathlibAnnex.Analysis.CStarAlgebra.IsCompactOperatorModel",
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  "conclusion": "π is injective and satisfies the full unbundled compact-operator model: every π(a) is compact and every compact T on H has an exact preimage in A.",
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  "definition_equation": "Injective π ∧ IsCompactOperatorModel π, i.e. Injective π ∧ (∀ a, IsCompactOperator (π a)) ∧ (∀ T, IsCompactOperator T → ∃ a, π a=T).",
  "definition_explanation": "Injective π ∧ IsCompactOperatorModel π, i.e. Injective π ∧ (∀ a, IsCompactOperator (π a)) ∧ (∀ T, IsCompactOperator T → ∃ a, π a=T).",
  "display_title": "Compact-operator model from a representative of the unique irreducible-representation class",
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  "name": "MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation.faithful_and_compactOperatorModel_of_singleton",
  "natural_language_statement": "Let A be a nonzero complex C*-algebra, not assumed unital, H a separable complex Hilbert space, and π a representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. No separability of A, prior faithfulness, or simplicity is assumed. π is injective and satisfies the full unbundled compact-operator model: every π(a) is compact and every compact T on H has an exact preimage in A.",
  "one_sentence_role": "Shows that a representative of the unique irreducible-representation class on a separable Hilbert space is faithful and has image exactly K(H).",
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  "proof_steps": [
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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 → Function.Injective ⇑pi ∧ MathlibAnnex.Analysis.CStarAlgebra.IsCompactOperatorModel pi"
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    {
      "step_id": "source_equation",
      "text": "Injective π ∧ IsCompactOperatorModel π, i.e. Injective π ∧ (∀ a, IsCompactOperator (π a)) ∧ (∀ T, IsCompactOperator T → ∃ a, π a=T)."
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      "step_id": "source_route",
      "text": "The source combines the faithfulness theorem for the unique-class representative with the compact-operator-model theorem. The latter gives compactness of every image and a preimage for every compact operator. This establishes the stated unbundled property; it does not construct a bundled StarAlgEquiv."
    }
  ],
  "proof_summary": "The source combines the faithfulness theorem for the unique-class representative with the compact-operator-model theorem. The latter gives compactness of every image and a preimage for every compact operator. This establishes the stated unbundled property; it does not construct a bundled StarAlgEquiv.",
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    "text": "theorem faithful_and_compactOperatorModel_of_singleton [Nontrivial A]\n    [TopologicalSpace.SeparableSpace H]\n    (pi : NonUnitalCStarRepresentation A H)\n    (hsingle : IsSingletonIrreducibleModel.{u, v, u} pi) :\n    Function.Injective pi ∧ IsCompactOperatorModel pi"
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