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  "assumptions": "Let A be a nonzero infinite-dimensional unital complex C*-algebra, H a separable complex Hilbert space, and π a unital representation of A on H.",
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      "label": "MathlibAnnex.Analysis.CStarAlgebra.Representation.finiteDimensional_algebra_of_singleton_amongNonUnital",
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  "conclusion": "The representation π is not a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. Equivalently, π is not both irreducible and unitarily equivalent to every nonzero irreducible representation of A. This does not rule out separable irreducible representations in general.",
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  "name": "MathlibAnnex.Analysis.CStarAlgebra.Representation.not_singleton_amongNonUnital_of_infiniteDimensional",
  "natural_language_statement": "Let A be a nonzero infinite-dimensional unital complex C*-algebra, H a separable complex Hilbert space, and π a unital representation of A on H. The representation π is not a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. Equivalently, π is not both irreducible and unitarily equivalent to every nonzero irreducible representation of A. This does not rule out separable irreducible representations in general.",
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      "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], ¬FiniteDimensional ℂ A → ∀ [TopologicalSpace.SeparableSpace H] (pi : MathlibAnnex.Analysis.CStarAlgebra.Representation A H), ¬pi.IsSingletonIrreducibleModelAmongNonUnital"
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      "text": "¬FiniteDimensional ℂ A → ∀ π on separable H, ¬IsSingletonIrreducibleModelAmongNonUnital π."
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      "text": "If π represented the unique irreducible-representation class of A, the preceding finite-dimensional-algebra theorem would force A to be finite-dimensional, contradicting the hypothesis."
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  "proof_summary": "If π represented the unique irreducible-representation class of A, the preceding finite-dimensional-algebra theorem would force A to be finite-dimensional, contradicting the hypothesis.",
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    "text": "theorem not_singleton_amongNonUnital_of_infiniteDimensional\n    [Nontrivial A] (hA : ¬ FiniteDimensional ℂ A)\n    [TopologicalSpace.SeparableSpace H]\n    (pi : Representation A H) :\n    ¬ IsSingletonIrreducibleModelAmongNonUnital.{u, v, u} pi"
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