nonUnitalCStarAlgebra — Mathlib

English

Let A be a nonunital C*-algebra and S be a closed star-subalgebra of A. Then S is a nonunital C*-algebra.

Русский

Пусть A — полная безедв C*-алгебра, и S — замкнутая звезд-подалгебра A. Тогда S является неединичной C*-алгеброй.

LaTeX

$$$NonUnitalCStarAlgebra(A) \wedge S \subseteq A \wedge IsClosed(S) \Rightarrow NonUnitalCStarAlgebra(S)$$$

Lean4

instance nonUnitalCStarAlgebra {S A : Type*} [NonUnitalCStarAlgebra A] [SetLike S A] [NonUnitalSubringClass S A]
    [SMulMemClass S ℂ A] [StarMemClass S A] (s : S) [h_closed : IsClosed (s : Set A)] : NonUnitalCStarAlgebra 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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