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  "name": "MathlibAnnex.Analysis.CStarAlgebra.Representation.finiteDimensional_space_of_singleton",
  "natural_language_statement": "Let A be a nonzero unital complex C*-algebra, 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. H is finite-dimensional over ℂ. The conclusion is conditional on the unique-class hypothesis, not a statement about all unital irreducible representations.",
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      "text": "The source first obtains injectivity and simplicity from the unique-class hypothesis for π, then a nonzero scalar-corner projection with compact image. The compact-preimage ideal is nonzero and closed, hence all of A by simplicity. Thus π(1)=id_H is compact, and the compact-identity criterion makes H finite-dimensional."
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    "text": "theorem finiteDimensional_space_of_singleton [Nontrivial A]\n    [TopologicalSpace.SeparableSpace H]\n    (pi : Representation A H)\n    (hsingle : IsSingletonIrreducibleModel.{u, v, u} pi) :\n    FiniteDimensional ℂ H"
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