"""Reproduce the original predual-smoothing diagram with no external library.""" from pathlib import Path import sys SVG = """ From quasi-equivalence to a continuous normal action For a nondegenerate representation on separable H, bounded sequential density makes scalar normal orbits Borel. Haar averages are normal by a faithful-state reduction from increasing nets to sequences. Left translation makes each average orbit norm continuous. Countably many separating tests on one finite-measure open set prove that the averages span a norm-dense subspace of the predual, which gives continuity for all normal functionals. Continuity comes from the whole predual Point-norm action α on A; nondegenerate π on a separable Hilbert space H. G arbitrary LCH; π ∘ αₛ is quasi-equivalent to π for every s ∈ G. QI1 · Bounded sequences give scalar measurability π(aₙ) → x strongly*, with ‖π(aₙ)‖ ≤ ‖x‖ ω(βₜ(x)) = limₙ ω(π(αₜ(aₙ))) Each approximating scalar orbit is continuous; the limit is Borel. QI2 · The Haar averages are normal Ω(f,ω)(x) = ∫ f(t) ω(βₜ⁻¹(x)) dt Tₛω = ω ∘ βₛ⁻¹ A faithful normal state selects xᵢₙ ↑ x from every bounded xᵢ ↑ x. Scalar monotone convergence + NF1–4 ⇒ Ω(f,ω) belongs to P∗. QI2 · Left translation gives norm continuity Tₛ Ω(f,ω) = Ω(Lₛf,ω) Lₛf(t) = f(s⁻¹t) ‖Tₛ Ω(f,ω) − Ω(f,ω)‖ ≤ ‖Lₛf − f‖₁ ‖ω‖ → 0 Left Haar measure is used throughout; no modular factor is omitted. QI3 · The averages span a norm-dense subspace of P∗ An annihilator x gives countably many scalar tests hₙ = 0 a.e. Use one open V with 0 < Haar(V) < ∞ to choose a common parameter. All normal tests then kill βₜ⁻¹(x), hence x = 0. ‖Tₛω − ω‖ ≤ 2‖ω − η‖ + ‖Tₛη − η‖ for averaged η P = π(A)″; P∗ is the predual. Arrows are proved implications. Mathematical antecedent: M. Takesaki, Theory of Operator Algebras II, X.3.1. """ def main(): target=Path(__file__).with_name("predual-smoothing.svg") data=SVG.encode("utf-8") if "--check" in sys.argv: if target.read_bytes()!=data: raise SystemExit("SVG differs from its reproducible source.") print("Exact SVG reproduction verified.") else: target.write_bytes(data) print(target.name) if __name__=="__main__": main()