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  "display_title": "A compact-operator model for a representation",
  "one_sentence_role": "Packages faithfulness together with the equality $e(A)=\\mathcal K(H)$.",
  "natural_language_statement": "Let $A$ be a complex $C^*$-algebra, with no unit assumed, let $H$ be a complex Hilbert space, and let $e:A\\to\\mathcal B(H)$ be a $*$-representation. The predicate below asserts that $e$ identifies $A$ exactly with the compact operators on $H$.",
  "definition_explanation": "The three conditions are $$e\\text{ is injective},\\qquad e(A)\\subseteq\\mathcal K(H),\\qquad\\mathcal K(H)\\subseteq e(A).$$ Equivalently, $e$ is faithful and $e(A)=\\mathcal K(H)$; hence $A$ is $*$-isomorphic to $\\mathcal K(H)$ through the specified representation.",
  "assumptions": "No separability, irreducibility or singleton-spectrum hypothesis is part of this definition. A unit is not assumed.",
  "conclusion": "The output is a property of the given representation $e$, not merely an assertion that some abstract $*$-isomorphism exists.",
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    "text": "def IsCompactOperatorModel [NonUnitalCStarAlgebra A]\n    (e : A →⋆ₙₐ[ℂ] (H →L[ℂ] H)) : Prop"
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