Inverses as limits of polynomials in Banach algebras

DefinitionScottish Book Problem 176 (Eidelheit's claim)#✓fully proved

The claim that in every complete normed ring \(A\) (so \(\|xy\| \le \|x\| \, \|y\|\), and \(A\) has a unit) that is a normed algebra over \(\mathbb{R}\), the inverse of every invertible element is a limit of real polynomials in that element: whenever \(a, b \in A\) satisfy \(ab = 1\) and \(ba = 1\), there is a sequence \((p_n)_{n \ge 0}\) of real polynomials with \(p_n(a) \to b\) as \(n \to \infty\).

def ScottishBookProblem176Claim : Prop :=
  ∀ (A : Type) [NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A]
    (a b : A), a * b = 1 → b * a = 1 →
    ∃ p : ℕ → Polynomial ℝ,
      Filter.Tendsto (fun n => Polynomial.aeval a (p n)) Filter.atTop (nhds b)

Used by 1

Used byEidelheit's Problem 176 has a negative answer

TheoremEidelheit's Problem 176 has a negative answer#●ready to prove

The claim of Scottish Book Problem 176 is false: there are a complete normed ring \(A\) that is a normed algebra over \(\mathbb{R}\) and elements \(a, b \in A\) with \(ab = 1\) and \(ba = 1\) such that \(b\) is not the limit of any sequence \((p_n(a))_{n \ge 0}\) with \(p_n\) real polynomials.

The counterexample recorded in the typescript is the ring of linear operations on the space \(C[0, 1]\), with \(a\) the operation \(U\) given by \(U(x)(t) = x(t^2)\) for \(0 \le t \le 1\).

theorem_wanted eidelheit176_refuted : ¬ ScottishBookProblem176Claim

Sources

• M. Eidelheit, Problem 176 of the Scottish Book, with the negative answer recorded in Ad 176 of the typescript (November 11, 1938). See the commentary in R. D. Mauldin (ed.), The Scottish Book, 2nd ed. (Birkhäuser, 2015).

Uses 1

Statement usesScottish Book Problem 176 (Eidelheit's claim)

1 fully proved1 ready to prove

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