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  "assumptions": "Let A be a nonzero complex C*-algebra, not assumed unital, 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.",
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      "label": "MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation.exists_nonzero_projection_scalar_corner",
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      "text": "Exact formal dependency; inspect the linked Card and exact source."
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  "conclusion": "There are a nonzero star projection p∈A and a unit vector e∈H with π(p)=rankOne e e; the represented operator π(p) is compact. Minimality is not an explicit clause of this theorem type.",
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  "definition_equation": "∃ p, IsStarProjection p ∧ p≠0 ∧ (∃ e, ‖e‖=1 ∧ π p=rankOne ℂ e e) ∧ IsCompactOperator (π p).",
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  "display_title": "A represented rank-one projection from a representative of the unique irreducible-representation class",
  "exact_elaborated_lean_type": "∀ {A : Type u} [inst : NonUnitalCStarAlgebra A] {H : Type v} [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℂ H] [inst_3 : CompleteSpace H] [Nontrivial A] [inst_5 : PartialOrder A] [StarOrderedRing A] [TopologicalSpace.SeparableSpace H] (pi : MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation A H), pi.IsSingletonIrreducibleModel → ∃ p, IsStarProjection p ∧ p ≠ 0 ∧ (∃ e, ‖e‖ = 1 ∧ pi p = ((InnerProductSpace.rankOne ℂ) e) e) ∧ IsCompactOperator ⇑(pi p)",
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  "name": "MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation.exists_nonzero_projection_rankOne_map",
  "natural_language_statement": "Let A be a nonzero complex C*-algebra, not assumed unital, 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. There are a nonzero star projection p∈A and a unit vector e∈H with π(p)=rankOne e e; the represented operator π(p) is compact. Minimality is not an explicit clause of this theorem type.",
  "one_sentence_role": "Produces a nonzero projection whose represented image is a compact rank-one orthogonal projection.",
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      "text": "Exact Lean statement: ∀ {A : Type u} [inst : NonUnitalCStarAlgebra A] {H : Type v} [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℂ H] [inst_3 : CompleteSpace H] [Nontrivial A] [inst_5 : PartialOrder A] [StarOrderedRing A] [TopologicalSpace.SeparableSpace H] (pi : MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation A H), pi.IsSingletonIrreducibleModel → ∃ p, IsStarProjection p ∧ p ≠ 0 ∧ (∃ e, ‖e‖ = 1 ∧ pi p = ((InnerProductSpace.rankOne ℂ) e) e) ∧ IsCompactOperator ⇑(pi p)"
    },
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      "text": "∃ p, IsStarProjection p ∧ p≠0 ∧ (∃ e, ‖e‖=1 ∧ π p=rankOne ℂ e e) ∧ IsCompactOperator (π p)."
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    {
      "step_id": "source_route",
      "text": "The preceding scalar-corner theorem gives p. Injectivity makes π(p) nonzero, and the scalar-corner/rank-one lemma gives a unit e and the image equation. Compactness follows from compactness of a rank-one operator."
    }
  ],
  "proof_summary": "The preceding scalar-corner theorem gives p. Injectivity makes π(p) nonzero, and the scalar-corner/rank-one lemma gives a unit e and the image equation. Compactness follows from compactness of a rank-one operator.",
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    "text": "theorem exists_nonzero_projection_rankOne_map [Nontrivial A]\n    [PartialOrder A] [StarOrderedRing A]\n    [TopologicalSpace.SeparableSpace H]\n    (pi : NonUnitalCStarRepresentation A H)\n    (hsingle : IsSingletonIrreducibleModel.{u, v, u} pi) :\n    ∃ p : A, IsStarProjection p ∧ p ≠ 0 ∧\n      (∃ e : H, ‖e‖ = 1 ∧ pi p = InnerProductSpace.rankOne ℂ e e) ∧\n      IsCompactOperator (pi p)"
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