cstarAlgebra — Mathlib

English

If S is a star-subalgebra of a C*-algebra A, closed as a subset, then S carries a C*-algebra structure as a subalgebra.

Русский

Если S — звездная подпольгебра (StarSubalgebra) под C*-алгеброй A и замкнута как множество, то S обладает структурой C*-алгебры как подалгебра.

LaTeX

$$$$\\text{cstarAlgebra}(S)\\quad\\text{for } S \\subseteq A \\text{ closed},$$$$

Lean4

instance cstarAlgebra {S A : Type*} [CStarAlgebra A] [SetLike S A] [SubringClass S A] [SMulMemClass S ℂ A]
    [StarMemClass S A] (s : S) [h_closed : IsClosed (s : Set A)] : CStarAlgebra s
    where
  toCompleteSpace := h_closed.completeSpace_coe
  norm_mul_self_le x := CStarRing.norm_star_mul_self (x := (x : A)) |>.symm.le

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