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Theorem. If B is a Banach algebra which is also a nil algebra, then B is a nilpotent algebra.Proof. For each positive integer j, define Nj = {xQB:x' = 0}.By the theorem of Nagata-Higman [l, p. 274], it will be enough to show that some N¡ = B. Each N¡ is closed and the union of the N¡ is P. Hence, by the Baire Category Theorem, there is a fixed integer k and a fixed zQB for which Nk is a neighborhood of z. Suppose xQB. Define the B valued polynomial of a scalar variable /, p(t) = (z+t(x -z))k. Since z+t(x -z) is continuous, p(t) must equal zero for all sufficiently small /. Therefore, p(t) = 0; and, in particular p(\) = x* = 0. 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.
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