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  "display_title": "Pure state detecting a nonzero square",
  "exact_elaborated_lean_type": "∀ {A : Type u} [inst : CStarAlgebra A] [inst_1 : PartialOrder A] [inst_2 : StarOrderedRing A] [Nontrivial A] {a : A}, a ≠ 0 → ∃ phi ∈ MathlibAnnex.Analysis.CStarAlgebra.stateSpace A, MathlibAnnex.Analysis.CStarAlgebra.IsPureState A phi ∧ phi (star a * a) ≠ 0",
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  "name": "MathlibAnnex.Analysis.CStarAlgebra.exists_pureState_nonzero_on_star_mul_self",
  "natural_language_statement": "Let A be a nonzero unital complex C*-algebra and let a ∈ A be nonzero. No separability assumption on A is required. Some pure state φ in stateSpace A satisfies φ(a* a)≠0.",
  "one_sentence_role": "Supplies a pure state that detects a* a, enabling faithfulness arguments without separability of the algebra.",
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      "text": "a ≠ 0 → ∃ φ ∈ stateSpace A, IsPureState A φ ∧ φ (star a * a) ≠ 0."
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      "text": "The source sets b=a* a, uses its nonzero spectral radius to obtain a nonzero character value on the closed elemental commutative star subalgebra generated by b, extends that character to a pure state, and transfers the nonzero value back to b."
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    "text": "theorem exists_pureState_nonzero_on_star_mul_self [Nontrivial A]\n    {a : A} (ha : a ≠ 0) :\n    ∃ phi : A →L[ℂ] ℂ,\n      phi ∈ stateSpace A ∧ IsPureState A phi ∧ phi (star a * a) ≠ 0"
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