range_cfc_nnreal_eq_image_cfc_real

English

IsClosed {a ∈ A | 0 ≤ a} in any nonunital C*-algebra A with star-ordered ring structure.

Русский

Замкнуто множество неотрицательных элементов {a ∈ A | 0 ≤ a} в любой безнулевой C*-алгебре A с звездно-упорядоченной структурой.

LaTeX

$$$\\mathrm{IsClosed}\\;\\{a \\in A \\mid 0 \\le a\\}$$$

Lean4

theorem range_cfc_nnreal_eq_image_cfc_real (a : A) (ha : 0 ≤ a) :
    Set.range (cfc (R := ℝ≥0) · a) = (cfc · a) '' {f | ∀ x ∈ spectrum ℝ a, 0 ≤ f x} :=
  by
  ext
  constructor
  · rintro ⟨f, rfl⟩
    simp only [cfc_nnreal_eq_real f a ha]
    exact ⟨_, fun _ _ ↦ by positivity, rfl⟩
  · rintro ⟨f, hf, rfl⟩
    simp only [cfc_real_eq_nnreal a hf]
    exact ⟨_, rfl⟩

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