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  "assumptions": "Let D be a unital star subalgebra of a unital complex C*-algebra A.",
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  "conclusion": "D is commutative, and any commutative unital star subalgebra E containing D is contained in D; thus D is maximal for inclusion among such subalgebras.",
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  "name": "MathlibAnnex.Analysis.CStarAlgebra.IsMaximalAbelian",
  "natural_language_statement": "Let D be a unital star subalgebra of a unital complex C*-algebra A. D is commutative, and any commutative unital star subalgebra E containing D is contained in D; thus D is maximal for inclusion among such subalgebras.",
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      "text": "IsMaximalAbelian D ↔ IsMulCommutative D ∧ ∀ E, IsMulCommutative E → D ≤ E → E ≤ D."
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      "text": "This definition does not itself add a closedness conjunct. Closedness, when needed later, comes from a separate theorem about maximal abelian subalgebras."
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    "text": "def IsMaximalAbelian (D : StarSubalgebra ℂ A) : Prop"
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