∀ {A : Type u_1} [inst : NonUnitalCStarAlgebra A] [inst_1 : PartialOrder A] {l : Filter A},
  (∀ (m : A), Filter.Tendsto (fun x => m * x) l (nhds m)) →
    (∀ (m : A), Filter.Tendsto (fun x => x * m) l (nhds m)) →
      ∀ [neBot : l.NeBot], (∀ᶠ (x : A) in l, 0 ≤ x) → (∀ᶠ (x : A) in l, ‖x‖ ≤ 1) → l.IsIncreasingApproximateUnit