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  "assumptions": "Let A be a nonzero unital complex C*-algebra, H a separable complex Hilbert space, and π a unital representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A.",
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  "conclusion": "π is injective; its underlying nonunital star homomorphism is injective, maps every a∈A to a compact operator, and represents every compact operator on H.",
  "declaration_kind": "theorem",
  "definition_equation": "Injective π ∧ IsCompactOperatorModel π.toNonUnitalStarAlgHom.",
  "definition_explanation": "Injective π ∧ IsCompactOperatorModel π.toNonUnitalStarAlgHom.",
  "display_title": "Compact-operator model from a unital representative of the unique irreducible-representation class",
  "exact_elaborated_lean_type": "∀ {A : Type u} [inst : CStarAlgebra 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.Representation A H), pi.IsSingletonIrreducibleModelAmongNonUnital → Function.Injective ⇑pi ∧ MathlibAnnex.Analysis.CStarAlgebra.IsCompactOperatorModel (StarAlgHom.toNonUnitalStarAlgHom pi)",
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  "name": "MathlibAnnex.Analysis.CStarAlgebra.Representation.faithful_and_compactOperatorModel_of_singleton_amongNonUnital",
  "natural_language_statement": "Let A be a nonzero unital complex C*-algebra, H a separable complex Hilbert space, and π a unital representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. π is injective; its underlying nonunital star homomorphism is injective, maps every a∈A to a compact operator, and represents every compact operator on H.",
  "one_sentence_role": "Shows that a separably acting unital representative of the unique irreducible-representation class is faithful and has image exactly K(H).",
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      "step_id": "formal_type",
      "text": "Exact Lean statement: ∀ {A : Type u} [inst : CStarAlgebra 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.Representation A H), pi.IsSingletonIrreducibleModelAmongNonUnital → Function.Injective ⇑pi ∧ MathlibAnnex.Analysis.CStarAlgebra.IsCompactOperatorModel (StarAlgHom.toNonUnitalStarAlgHom pi)"
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      "text": "Injective π ∧ IsCompactOperatorModel π.toNonUnitalStarAlgHom."
    },
    {
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      "text": "The source passes from the nonunital-interface formulation of the unique-class condition to its unital formulation and applies the unital faithful-and-compact-model theorem. The model entails injectivity, compactness of each image and a preimage for every compact T; no bundled StarAlgEquiv is produced."
    }
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  "proof_summary": "The source passes from the nonunital-interface formulation of the unique-class condition to its unital formulation and applies the unital faithful-and-compact-model theorem. The model entails injectivity, compactness of each image and a preimage for every compact T; no bundled StarAlgEquiv is produced.",
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    "text": "theorem faithful_and_compactOperatorModel_of_singleton_amongNonUnital\n    [Nontrivial A] [TopologicalSpace.SeparableSpace H]\n    (pi : Representation A H)\n    (hsingle : IsSingletonIrreducibleModelAmongNonUnital.{u, v, u} pi) :\n    Function.Injective pi ∧\n      IsCompactOperatorModel pi.toNonUnitalStarAlgHom"
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