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  "assumptions": "Let A be a nonzero complex C*-algebra, not assumed unital, H a 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 assumption on H is required.",
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  "conclusion": "Then π is injective.",
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  "name": "MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation.injective_of_singleton",
  "natural_language_statement": "Let A be a nonzero complex C*-algebra, not assumed unital, H a 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 assumption on H is required. Then π is injective.",
  "one_sentence_role": "Shows that a representative of the unique irreducible-representation class of a nonzero C*-algebra is faithful.",
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      "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] (pi : MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation A H), pi.IsSingletonIrreducibleModel → Function.Injective ⇑pi"
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      "text": "IsSingletonIrreducibleModel.{u,v,u} π → Injective π."
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      "text": "For π(x−y)=0, the source assumes x−y≠0, embeds it into the minimal unitization, and finds a pure state detecting its star-square. Its GNS representation restricts to a nonzero irreducible representation of A. The unique-class hypothesis gives a unitary equivalence to π, which would force the detected element to act as zero there, a contradiction."
    }
  ],
  "proof_summary": "For π(x−y)=0, the source assumes x−y≠0, embeds it into the minimal unitization, and finds a pure state detecting its star-square. Its GNS representation restricts to a nonzero irreducible representation of A. The unique-class hypothesis gives a unitary equivalence to π, which would force the detected element to act as zero there, a contradiction.",
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    "start_byte": 1239,
    "text": "theorem injective_of_singleton [Nontrivial A]\n    (pi : NonUnitalCStarRepresentation A H)\n    (hsingle : IsSingletonIrreducibleModel.{u, v, u} pi) :\n    Function.Injective pi"
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