The nilpotency of Banach nil algebras

Abstract

By the theorem of Nagata-Higman [1, p. 274], it will be enough to show that some Nj = B. Each Nj is closed and the union of the N1 is B. Hence, by the Baire Category Theorem, there is a fixed integer k and a fixed zEB for which Nk is a neighborhood of z. Suppose xEB. Define the B valued polynomial of a scalar variable t, p(t) = (z+t(x-z))k. Since z+t(x-z) is continuous, p(t) must equal zero for all sufficiently small t. Therefore, p(t) 0; and, in particular p(1) =xk=O. This completes the proof. It is easy to construct examples of normed algebras for which the above theorem is false, though the theorem is true for second category topological algebras.