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  "assumptions": "Let A be a complex C*-algebra, not assumed unital, and H a complex Hilbert space.",
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  "conclusion": "A representation of A on H is a complex-linear nonunital star-algebra homomorphism into the bounded complex-linear operators on H. It preserves addition, multiplication, complex scalar multiplication and involution; neither preservation of a unit nor faithfulness is built in.",
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  "definition_equation": "NonUnitalCStarRepresentation A H = A →⋆ₙₐ[ℂ] (H →L[ℂ] H).",
  "definition_explanation": "NonUnitalCStarRepresentation A H = A →⋆ₙₐ[ℂ] (H →L[ℂ] H).",
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  "name": "MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation",
  "natural_language_statement": "Let A be a complex C*-algebra, not assumed unital, and H a complex Hilbert space. A representation of A on H is a complex-linear nonunital star-algebra homomorphism into the bounded complex-linear operators on H. It preserves addition, multiplication, complex scalar multiplication and involution; neither preservation of a unit nor faithfulness is built in.",
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      "text": "Exact Lean statement: (A : Type u) → [NonUnitalCStarAlgebra A] → (H : Type v) → [inst : NormedAddCommGroup H] → [InnerProductSpace ℂ H] → [CompleteSpace H] → Type (max u v)"
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      "text": "NonUnitalCStarRepresentation A H = A →⋆ₙₐ[ℂ] (H →L[ℂ] H)."
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    {
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      "text": "This is a type abbreviation. Unfolding it gives the nonunital star-homomorphism type; later theorems separately assume nonzero action, irreducibility or injectivity."
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    "text": "abbrev NonUnitalCStarRepresentation (A : Type u)\n    [NonUnitalCStarAlgebra A] (H : Type v)\n    [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]"
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