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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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  "name": "MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation.exists_nonzero_projection_scalar_corner",
  "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 is a nonzero star projection p in A whose every corner p a p is a complex scalar multiple of p.",
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      "text": "Exact Lean statement: ∀ {A : Type u} [inst : NonUnitalCStarAlgebra A] [inst_1 : PartialOrder A] [StarOrderedRing A] {H : Type v} [inst_3 : NormedAddCommGroup H] [inst_4 : InnerProductSpace ℂ H] [inst_5 : CompleteSpace H] [Nontrivial A] [TopologicalSpace.SeparableSpace H] (pi : MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation A H), pi.IsSingletonIrreducibleModel → ∃ p, IsStarProjection p ∧ p ≠ 0 ∧ ∀ (a : A), ∃ c, p * a * p = c • p"
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      "text": "∃ p : A, IsStarProjection p ∧ p≠0 ∧ ∀ a : A, ∃ c : ℂ, p*a*p=c•p."
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      "text": "The source takes a nonzero a, inserts b=a* a into a maximal abelian subalgebra of the unitization, uses separability to count its characters, and obtains an isolated character distinct from the scalar character. A projection supported at that character has a scalar corner; the zero scalar coordinate forces that projection back into A."
    }
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  "proof_summary": "The source takes a nonzero a, inserts b=a* a into a maximal abelian subalgebra of the unitization, uses separability to count its characters, and obtains an isolated character distinct from the scalar character. A projection supported at that character has a scalar corner; the zero scalar coordinate forces that projection back into A.",
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    "text": "theorem exists_nonzero_projection_scalar_corner [Nontrivial 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      ∀ a : A, ∃ c : ℂ, p * a * p = c • p"
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