tendsto_mul_left_iff_tendsto_mul_right

English

The family of sections given by nonneg elements with norms less than 1 forms a basis for the nonunital C*-algebra's topology.

Русский

Семейство секций, заданных неотрицательными элементами с нормой меньшей единицы, образует базис топологии нелокальной C*-алгебры.

LaTeX

$$$$ \text{isBasis}_{A} = \{ x \in A \mid 0 \le x, \|x\| < 1 \} $$$$

Lean4

/-- Multiplication on the left by `m` tends to `𝓝 m` if and only if multiplication on the right
does, provided the elements are eventually selfadjoint along the filter `l`. -/
theorem tendsto_mul_left_iff_tendsto_mul_right {l : Filter A} (hl : ∀ᶠ x in l, IsSelfAdjoint x) :
    (∀ m, Tendsto (m * ·) l (𝓝 m)) ↔ (∀ m, Tendsto (· * m) l (𝓝 m)) :=
  by
  refine ⟨fun h m ↦ ?_, fun h m ↦ ?_⟩
  all_goals
    apply (star_star m ▸ (continuous_star.tendsto _ |>.comp <| h (star m))).congr'
    filter_upwards [hl] with x hx
    simp [hx.star_eq]

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